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== 4.1#h(0.6em)Use the Rectangular Coordinate System

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Evaluate #math.equation(block: false, alt: "x plus 3")[$x + 3$] when #math.equation(block: false, alt: "x equals −1")[$x = −1$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "2 x minus 5 y")[$2 x − 5 y$] when #math.equation(block: false, alt: "x equals 3")[$x = 3$] and #math.equation(block: false, alt: "y equals −2 .")[$y = −2 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
16
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve for #math.equation(block: false, alt: "y")[$y$]: #math.equation(block: false, alt: "40 minus 4 y equals 20")[$40 − 4 y = 20$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
5
]
]


=== Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a #strong[rectangular coordinate system]. The rectangular coordinate system is also called the #emph[xy]-plane or the ‘coordinate plane’.

The horizontal number line is called the #emph[x-axis]. The vertical number line is called the #emph[y-axis.] The #emph[x]-axis and the #emph[y]-axis together form the rectangular coordinate system. These axes divide a plane into four regions, called #strong[quadrants]. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The top-right portion of the plane is labeled "I", the top-left portion of the plane is labeled "II", the bottom-left portion of the plane is labeled "III" and the bottom-right portion of the plane is labeled "IV".], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The top-right portion of the plane is labeled \"I\", the top-left portion of the plane is labeled \"II\", the bottom-left portion of the plane is labeled \"III\" and the bottom-right portion of the plane is labeled \"IV\".", caption: [‘Quadrant’ has the root ‘quad,’ which means ‘four.’])

In the #strong[rectangular coordinate system], every point is represented by an #emph[ordered pair]. The first number in the ordered pair is the #strong[#emph[x]-coordinate] of the point, and the second number is the #strong[#emph[y]-coordinate] of the point.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Ordered Pair]

An #strong[ordered pair], #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$], gives the coordinates of a point in a rectangular coordinate system.

#figure(figph[The ordered pair x y is labeled with the first coordinate x labeled as "x-coordinate" and the second coordinate y labeled as "y-coordinate".], alt: "The ordered pair x y is labeled with the first coordinate x labeled as \"x-coordinate\" and the second coordinate y labeled as \"y-coordinate\".", caption: none)

The first number is the #emph[x]-coordinate.

The second number is the #emph[y]-coordinate.
]

The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]. The point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] has a special name. It is called the #strong[origin].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[The Origin]

The point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is called the #strong[origin]. It is the point where the #emph[x]-axis and #emph[y]-axis intersect.
]

We use the coordinates to locate a point on the #emph[xy]-plane. Let’s plot the point #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$] as an example. First, locate 1 on the #emph[x]-axis and lightly sketch a vertical line through #math.equation(block: false, alt: "x equals 1")[$x = 1$]. Then, locate 3 on the #emph[y]-axis and sketch a horizontal line through #math.equation(block: false, alt: "y equals 3")[$y = 3$]. Now, find the point where these two lines meet—that is the point with coordinates #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$].

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. The point (1, 3) is plotted and labeled. Two dotted lines, one parallel to the x-axis, the other parallel to the y-axis, meet perpendicularly at 1, 3. The dotted line parallel to the x-axis intercepts the y-axis at 3. The dotted line parallel to the y-axis intercepts the x-axis at 1.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. The point (1, 3) is plotted and labeled. Two dotted lines, one parallel to the x-axis, the other parallel to the y-axis, meet perpendicularly at 1, 3. The dotted line parallel to the x-axis intercepts the y-axis at 3. The dotted line parallel to the y-axis intercepts the x-axis at 1.", caption: none)

Notice that the vertical line through #math.equation(block: false, alt: "x equals 1")[$x = 1$] and the horizontal line through #math.equation(block: false, alt: "y equals 3")[$y = 3$] are not part of the graph. We just used them to help us locate the point #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$].

#examplebox("Example 1")[][
Plot each point in the rectangular coordinate system and identify the #strong[quadrant] in which the point is located:

ⓐ #math.equation(block: false, alt: "open parenthesis −5 , 4 close parenthesis")[$( −5 , 4 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −2 , 3 close parenthesis")[$( −2 , 3 )$] ⓔ #math.equation(block: false, alt: "open parenthesis 3 , the fraction 5 over 2 close parenthesis")[$( 3 , frac(5, 2) )$].

#solutionbox[
The first number of the coordinate pair is the #emph[x]-coordinate, and the second number is the #emph[y]-coordinate.

+ ⓐ Since #math.equation(block: false, alt: "x equals −5")[$x = −5$], the point is to the left of the #emph[y]-axis. Also, since #math.equation(block: false, alt: "y equals 4")[$y = 4$], the point is above the #emph[x]-axis. The point #math.equation(block: false, alt: "open parenthesis −5 , 4 close parenthesis")[$( −5 , 4 )$] is in Quadrant II.
+ ⓑ Since #math.equation(block: false, alt: "x equals −3")[$x = −3$], the point is to the left of the #emph[y]-axis. Also, since #math.equation(block: false, alt: "y equals −4")[$y = −4$], the point is below the #emph[x]-axis. The point #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] is in Quadrant III.
+ ⓒ Since #math.equation(block: false, alt: "x equals 2")[$x = 2$], the point is to the right of the #emph[y]-axis. Since #math.equation(block: false, alt: "y equals −3")[$y = −3$], the point is below the #emph[x]-axis. The point #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$] is in Quadrant lV.
+ ⓓ Since #math.equation(block: false, alt: "x equals −2")[$x = −2$], the point is to the left of the #emph[y]-axis. Since #math.equation(block: false, alt: "y equals 3")[$y = 3$], the point is above the #emph[x]-axis. The point #math.equation(block: false, alt: "open parenthesis −2 , 3 close parenthesis")[$( −2 , 3 )$] is in Quadrant II.
+ ⓔ Since #math.equation(block: false, alt: "x equals 3")[$x = 3$], the point is to the right of the #emph[y]-axis. Since #math.equation(block: false, alt: "y equals the fraction 5 over 2")[$y = frac(5, 2)$], the point is above the #emph[x]-axis. (It may be helpful to write #math.equation(block: false, alt: "the fraction 5 over 2")[$frac(5, 2)$] as a mixed number or decimal.) The point #math.equation(block: false, alt: "open parenthesis 3 , the fraction 5 over 2 close parenthesis")[$( 3 , frac(5, 2) )$] is in Quadrant I. #linebreak() 
#figure(figph[This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The following points are labeled: (3, 5 divided by 2), negative 5, 4), (negative 3, negative 4), (0, negative 1), and (2, negative 3).], alt: "This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 6 to 6. The following points are labeled: (3, 5 divided by 2), negative 5, 4), (negative 3, negative 4), (0, negative 1), and (2, negative 3).", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:

ⓐ #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −4 close parenthesis")[$( 4 , −4 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −4 , 4 close parenthesis")[$( −4 , 4 )$] ⓔ #math.equation(block: false, alt: "open parenthesis −4 , the fraction 3 over 2 close parenthesis")[$( −4 , frac(3, 2) )$].

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 2, 1) is plotted and labeled "a". The point (negative 3, negative 1) is plotted and labeled "b". The point (4, negative 4) is plotted and labeled "c". The point (negative 4, negative one half) is plotted and labeled “d”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 2, 1) is plotted and labeled \"a\". The point (negative 3, negative 1) is plotted and labeled \"b\". The point (4, negative 4) is plotted and labeled \"c\". The point (negative 4, negative one half) is plotted and labeled “d”.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:

ⓐ #math.equation(block: false, alt: "open parenthesis −4 , 1 close parenthesis")[$( −4 , 1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −2 , 3 close parenthesis")[$( −2 , 3 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 2 , −5 close parenthesis")[$( 2 , −5 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −2 , 5 close parenthesis")[$( −2 , 5 )$] ⓔ #math.equation(block: false, alt: "open parenthesis −3 , the fraction 5 over 2 close parenthesis")[$( −3 , frac(5, 2) )$].

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled "a". The point (negative 2, 3) is plotted and labeled "b". The point (2, negative 5) is plotted and labeled "c". The point (negative 3, 2 and one half) is plotted and labeled “d”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled \"a\". The point (negative 2, 3) is plotted and labeled \"b\". The point (2, negative 5) is plotted and labeled \"c\". The point (negative 3, 2 and one half) is plotted and labeled “d”.", caption: none)
]
]

How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.

For the point in  in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of points in the third quadrant? The second quadrant? The first quadrant?

Can you tell just by looking at the coordinates in which quadrant the point #math.equation(block: false, alt: "open parenthesis −2 , 5 close parenthesis")[$( −2 , 5 )$] is located? In which quadrant is #math.equation(block: false, alt: "open parenthesis 2 , −5 close parenthesis")[$( 2 , −5 )$] located?

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Quadrants]

We can summarize sign patterns of the quadrants in this way.

#math.equation(block: true, alt: "Quadrant I, Quadrant II, Quadrant III, Quadrant IV; open parenthesis x , y close parenthesis, open parenthesis x , y close parenthesis, open parenthesis x , y close parenthesis, open parenthesis x , y close parenthesis; open parenthesis plus , plus close parenthesis, open parenthesis − , plus close parenthesis, open parenthesis − , − close parenthesis, open parenthesis plus , − close parenthesis")[$"Quadrant I" &  &  & "Quadrant II" &  &  & "Quadrant III" &  &  & "Quadrant IV" \ ( x , y ) &  &  & ( x , y ) &  &  & ( x , y ) &  &  & ( x , y ) \ ( + , + ) &  &  & ( "−" , + ) &  &  & ( "−" , "−" ) &  &  & ( + , "−" )$]#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled "I" and "ordered pair +, +", the top-left portion of the plane is labeled "II" and "ordered pair -, +", the bottom-left portion of the plane is labelled "III" "ordered pair -, -" and the bottom-right portion of the plane is labeled "IV" and "ordered pair +, -".], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled \"I\" and \"ordered pair +, +\", the top-left portion of the plane is labeled \"II\" and \"ordered pair -, +\", the bottom-left portion of the plane is labelled \"III\" \"ordered pair -, -\" and the bottom-right portion of the plane is labeled \"IV\" and \"ordered pair +, -\".", caption: none)
]

What if one coordinate is zero as shown in ? Where is the point #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] located? Where is the point #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] located?

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. Points (0, 4) and (negative 2, 0) are plotted and labeled.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. Points (0, 4) and (negative 2, 0) are plotted and labeled.", caption: none)

The point #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] is on the #emph[y]-axis and the point #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] is on the #emph[x]-axis.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Points on the Axes]

Points with a #strong[#emph[y]-coordinate] equal to 0 are on the #emph[x]-axis, and have coordinates #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$].

Points with an #strong[#emph[x]-coordinate] equal to 0 are on the #emph[y]-axis, and have coordinates #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis")[$( 0 , b )$].
]

#examplebox("Example 2")[][
Plot each point:

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓔ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$].

#solutionbox[
+ ⓐ Since #math.equation(block: false, alt: "x equals 0")[$x = 0$], the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] is on the #emph[y]-axis.
+ ⓑ Since #math.equation(block: false, alt: "y equals 0")[$y = 0$], the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$] is on the #emph[x]-axis.
+ ⓒ Since #math.equation(block: false, alt: "y equals 0")[$y = 0$], the point whose coordinates are #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] is on the #emph[x]-axis.
+ ⓓ Since #math.equation(block: false, alt: "x equals 0")[$x = 0$] and #math.equation(block: false, alt: "y equals 0")[$y = 0$], the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] is the origin.
+ ⓔ Since #math.equation(block: false, alt: "x equals 0")[$x = 0$], the point whose coordinates are #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] is on the #emph[y]-axis. #linebreak() 
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The points (negative 3, 0), (0, 0), (0, negative 1), (0, 5), and (4, 0) are plotted and labeled.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The points (negative 3, 0), (0, 0), (0, negative 1), (0, 5), and (4, 0) are plotted and labeled.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Plot each point:

ⓐ #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] ⓔ #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$].

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Plot each point:

ⓐ #math.equation(block: false, alt: "open parenthesis −5 , 0 close parenthesis")[$( −5 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$].

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (negative 5, 0), (3, 0), (0, 0), (0, negative 1), and (0, 4) are plotted and labeled.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (negative 5, 0), (3, 0), (0, 0), (0, negative 1), and (0, 4) are plotted and labeled.", caption: none)
]
]

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the #emph[x]-coordinate of a point on a graph, read the number on the #emph[x]-axis directly above or below the point. To identify the #emph[y]-coordinate of a point, read the number on the #emph[y]-axis directly to the left or right of the point. Remember, when you write the #strong[ordered pair] use the correct order, #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$].

#examplebox("Example 3")[][
Name the ordered pair of each point shown in the rectangular coordinate system.

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled A, B, C, D, and E, respectively.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled A, B, C, D, and E, respectively.", caption: none)

#solutionbox[
Point A is above #math.equation(block: false, alt: "−3")[$−3$] on the #emph[x]-axis, so the #emph[x]-coordinate of the point is #math.equation(block: false, alt: "−3")[$−3$].

- The point is to the left of 3 on the #emph[y]-axis, so the #emph[y]-coordinate of the point is 3.
- The coordinates of the point are #math.equation(block: false, alt: "open parenthesis −3 , 3 close parenthesis")[$( −3 , 3 )$].

Point B is below #math.equation(block: false, alt: "−1")[$−1$] on the #emph[x]-axis, so the #emph[x]-coordinate of the point is #math.equation(block: false, alt: "−1")[$−1$].

- The point is to the left of #math.equation(block: false, alt: "−3")[$−3$] on the #emph[y]-axis, so the #emph[y]-coordinate of the point is #math.equation(block: false, alt: "−3")[$−3$].
- The coordinates of the point are #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$].

Point C is above 2 on the #emph[x]-axis, so the #emph[x]-coordinate of the point is 2.

- The point is to the right of 4 on the #emph[y]-axis, so the #emph[y]-coordinate of the point is 4.
- The coordinates of the point are #math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$].

Point D is below 4 on the #emph[x]-axis, so the #emph[x]-coordinate of the point is 4.

- The point is to the right of #math.equation(block: false, alt: "−4")[$−4$] on the #emph[y]-axis, so the #emph[y]-coordinate of the point is #math.equation(block: false, alt: "−4 .")[$−4 .$]
- The coordinates of the point are #math.equation(block: false, alt: "open parenthesis 4 , −4 close parenthesis")[$( 4 , −4 )$].

Point E is on the #emph[y]-axis at #math.equation(block: false, alt: "y equals −2")[$y = −2$]. The coordinates of point E are #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis .")[$( 0 , −2 ) .$]

Point F is on the #emph[x]-axis at #math.equation(block: false, alt: "x equals 3")[$x = 3$]. The coordinates of point F are #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis .")[$( 3 , 0 ) .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Name the ordered pair of each point shown in the rectangular coordinate system.

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled A, B, C, D, and E, respectively.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (4, 0), (negative 2, 0), (0, 0), (0, 2), and (0, negative 3) are plotted and labeled A, B, C, D, and E, respectively.", caption: none)

#solutionbox[
A: #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] B: #math.equation(block: false, alt: "open parenthesis −2 , 4 close parenthesis")[$( −2 , 4 )$] C: #math.equation(block: false, alt: "open parenthesis −5 , −1 close parenthesis")[$( −5 , −1 )$] D: #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$( 3 , −2 )$] E: #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$] F: #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Name the ordered pair of each point shown in the rectangular coordinate system.

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (negative 5, 0), (3, 0), (0, 0), (0, negative 1), and (0, 4) are plotted and labeled A, B, C, D, and E, respectively.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The points (negative 5, 0), (3, 0), (0, 0), (0, negative 1), and (0, 4) are plotted and labeled A, B, C, D, and E, respectively.", caption: none)

#solutionbox[
A: #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis")[$( 4 , 2 )$] B: #math.equation(block: false, alt: "open parenthesis −2 , 3 close parenthesis")[$( −2 , 3 )$] C: #math.equation(block: false, alt: "open parenthesis −4 , −4 close parenthesis")[$( −4 , −4 )$] D: #math.equation(block: false, alt: "open parenthesis 3 , −5 close parenthesis")[$( 3 , −5 )$] E: #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] F: #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]
]
]


=== Verify Solutions to an Equation in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like #math.equation(block: false, alt: "x equals 4")[$x = 4$]. (Then, you checked the solution by substituting back into the equation.)

Here’s an example of an equation in one variable, and its one solution.

#math.equation(block: true, alt: "3 x plus 5, equals, 17; 3 x, equals, 12; x, equals, 4")[$3 x + 5 & = & 17 \ 3 x & = & 12 \ x & = & 4$]But equations can have more than one variable. Equations with two variables may be of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$]. Equations of this form are called #strong[linear equations in two variables].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Linear Equation]

An equation of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$], where #math.equation(block: false, alt: "A")[$A$] and #math.equation(block: false, alt: "B")[$B$] are not both zero, is called a #strong[linear equation] #strong[in two variables].
]

Notice the word #emph[line] in #strong[linear]. Here is an example of a linear equation in two variables, #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$].

#figure(figph[In this figure, we see the linear equation Ax plus By equals C. Below this is the equation x plus 4y equals 8. Below this are the values A equals 1, B equals 4, and C equals 8.], alt: "In this figure, we see the linear equation Ax plus By equals C. Below this is the equation x plus 4y equals 8. Below this are the values A equals 1, B equals 4, and C equals 8.", caption: none)

The equation #math.equation(block: false, alt: "y equals −3 x plus 5")[$y = −3 x + 5$] is also a #strong[linear equation]. But it does not appear to be in the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$]. We can use the Addition Property of Equality and rewrite it in #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$] form.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y equals −3 x plus 5")[$#h(2.0em) y #h(0.5em) = #h(0.5em) −3 x + 5$]]),
  [Add to both sides.], [#math.equation(block: false, alt: "y plus 3 x equals −3 x plus 5 plus 3 x")[$y + 3 x #h(0.5em) = #h(0.5em) −3 x + 5 + 3 x$]],
  [Simplify.], [#math.equation(block: false, alt: "y plus 3 x equals 5")[$y + 3 x #h(0.5em) = #h(0.5em) 5$]],
  [Use the Commutative Property to put it in #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$] form.], [#math.equation(block: false, alt: "3 x plus y equals 5")[$3 x + y #h(0.5em) = #h(0.5em) 5$]],
))

By rewriting #math.equation(block: false, alt: "y equals −3 x plus 5")[$y = −3 x + 5$] as #math.equation(block: false, alt: "3 x plus y equals 5")[$3 x + y = 5$], we can easily see that it is a linear equation in two variables because it is of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$]. When an equation is in the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$], we say it is in #emph[standard form].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Standard Form of Linear Equation]

A linear equation is in standard form when it is written #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$].
]

Most people prefer to have #math.equation(block: false, alt: "A")[$A$], #math.equation(block: false, alt: "B")[$B$], and #math.equation(block: false, alt: "C")[$C$] be integers and #math.equation(block: false, alt: "A greater than or equal to 0")[$A ≥ 0$] when writing a linear equation in standard form, although it is not strictly necessary.

Linear equations have infinitely many solutions. For every number that is substituted for #math.equation(block: false, alt: "x")[$x$] there is a corresponding #math.equation(block: false, alt: "y")[$y$] value. This pair of values is a #emph[solution] to the linear equation and is represented by the ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]. When we substitute these values of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solution of a Linear Equation in Two Variables]

An #strong[ordered pair] #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] is a #strong[solution] of the linear equation #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$], if the equation is a true statement when the #emph[x]- and #emph[y]-values of the ordered pair are substituted into the equation.
]

#examplebox("Example 4")[][
Determine which ordered pairs are solutions to the equation #math.equation(block: false, alt: "x plus 4 y equals 8")[$x + 4 y = 8$].

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 , −4 close parenthesis")[$( 2 , −4 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$( −4 , 3 )$]

#solutionbox[
Substitute the #emph[x]- and #emph[y]-values from each ordered pair into the equation and determine if the result is a true statement. #linebreak() 
#figure(figph[The equation x plus 4y equals 8 is checked for three ordered pairs. The pair (0, 2) satisfies the equation; substituting gives 8 equals 8. The pair (2, negative 4) does not satisfy the equation; substituting gives negative 14 does not equal 8. The pair (negative 4, 3) satisfies the equation; substituting gives 8 equals 8.], alt: "The equation x plus 4y equals 8 is checked for three ordered pairs. The pair (0, 2) satisfies the equation; substituting gives 8 equals 8. The pair (2, negative 4) does not satisfy the equation; substituting gives negative 14 does not equal 8. The pair (negative 4, 3) satisfies the equation; substituting gives 8 equals 8.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following ordered pairs are solutions to #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$]?

ⓐ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 6 , −2 close parenthesis")[$( 6 , −2 )$]

#solutionbox[
a, c
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following ordered pairs are solutions to the equation #math.equation(block: false, alt: "4 x minus y equals 8")[$4 x − y = 8$]?

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 8 close parenthesis")[$( 0 , 8 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 1 , −4 close parenthesis")[$( 1 , −4 )$]

#solutionbox[
b, c
]
]

#examplebox("Example 5")[][
Which of the following ordered pairs are solutions to the equation #math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]?

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −7 close parenthesis")[$( −2 , −7 )$]

#solutionbox[
Substitute the #emph[x]- and #emph[y]-values from each #strong[ordered pair] into the equation and determine if it results in a true statement. #linebreak() 
#figure(figph[The graph shows three ordered pairs and checks if they are solutions to y equals 5x minus 1. The first pair (0, -1) satisfies the equation. The second pair (1, 4) also satisfies the equation. The third pair (-2, -7) does not satisfy the equation.], alt: "The graph shows three ordered pairs and checks if they are solutions to y equals 5x minus 1. The first pair (0, -1) satisfies the equation. The second pair (1, 4) also satisfies the equation. The third pair (-2, -7) does not satisfy the equation.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following ordered pairs are solutions to the equation #math.equation(block: false, alt: "y equals 4 x minus 3")[$y = 4 x − 3$]?

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −1 , −1 close parenthesis")[$( −1 , −1 )$]

#solutionbox[
b
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Which of the following ordered pairs are solutions to the equation #math.equation(block: false, alt: "y equals −2 x plus 6")[$y = −2 x + 6$]?

ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 6 close parenthesis")[$( 0 , 6 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$] ⓒ #math.equation(block: false, alt: "open parenthesis −2 , −2 close parenthesis")[$( −2 , −2 )$]

#solutionbox[
a, b
]
]


=== Complete a Table of Solutions to a Linear Equation in Two Variables

In the examples above, we substituted the #emph[x]- and #emph[y]-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for #math.equation(block: false, alt: "x")[$x$] and then solve the equation for #math.equation(block: false, alt: "y")[$y$]. Or, pick a value for #math.equation(block: false, alt: "y")[$y$] and then solve for #math.equation(block: false, alt: "x")[$x$].

We’ll start by looking at the solutions to the equation #math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$] that we found in . We can summarize this information in a table of solutions, as shown.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]],
  [1], [4], [#math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]],
))

To find a third solution, we’ll let #math.equation(block: false, alt: "x equals 2")[$x = 2$] and solve for #math.equation(block: false, alt: "y")[$y$].

#figure(figph[The figure shows the steps to solve for y when x equals 2 in the equation y equals 5 x minus 1. The equation y equals 5 x minus 1 is shown. Below it is the equation with 2 substituted in for x which is y equals 5 times 2 minus 1. To solve for y first multiply so that the equation becomes y equals 10 minus 1 then subtract so that the equation is y equals 9.], alt: "The figure shows the steps to solve for y when x equals 2 in the equation y equals 5 x minus 1. The equation y equals 5 x minus 1 is shown. Below it is the equation with 2 substituted in for x which is y equals 5 times 2 minus 1. To solve for y first multiply so that the equation becomes y equals 10 minus 1 then subtract so that the equation is y equals 9.", caption: none)

The ordered pair #math.equation(block: false, alt: "open parenthesis 2 , 9 close parenthesis")[$( 2 , 9 )$] is a solution to #math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]. We will add it to .

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 5 x minus 1")[$y = 5 x − 1$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]],
  [1], [4], [#math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]],
  [2], [9], [#math.equation(block: false, alt: "open parenthesis 2 , 9 close parenthesis")[$( 2 , 9 )$]],
))

We can find more solutions to the equation by substituting in any value of #math.equation(block: false, alt: "x")[$x$] or any value of #math.equation(block: false, alt: "y")[$y$] and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.

#examplebox("Example 6")[][
Complete  to find three solutions to the equation #math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$].

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [], [],
  [#math.equation(block: false, alt: "−1")[$−1$]], [], [],
  [2], [], [],
))

#solutionbox[
Substitute #math.equation(block: false, alt: "x equals 0")[$x = 0$], #math.equation(block: false, alt: "x equals −1")[$x = −1$], and #math.equation(block: false, alt: "x equals 2")[$x = 2$] into #math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]. #linebreak() 
#figure(figph[Table showing three points on the line y equals 4x minus 2. The first point is x equals 0, y equals negative 2, or the ordered pair 0, negative 2. The second point is x equals negative 1, y equals negative 6, or the ordered pair negative 1, negative 6. The third point is x equals 2, y equals 6, or the ordered pair 2, 6.], alt: "Table showing three points on the line y equals 4x minus 2. The first point is x equals 0, y equals negative 2, or the ordered pair 0, negative 2. The second point is x equals negative 1, y equals negative 6, or the ordered pair negative 1, negative 6. The third point is x equals 2, y equals 6, or the ordered pair 2, 6.", caption: none)

The results are summarized in .

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 4 x minus 2")[$y = 4 x − 2$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]],
  [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −1 , −6 close parenthesis")[$( −1 , −6 )$]],
  [2], [6], [#math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Complete the table to find three solutions to this equation: #math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$].

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [], [],
  [#math.equation(block: false, alt: "−1")[$−1$]], [], [],
  [2], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 3 x minus 1")[$y = 3 x − 1$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]],
  [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis")[$( −1 , −4 )$]],
  [2], [5], [#math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Complete the table to find three solutions to this equation: #math.equation(block: false, alt: "y equals 6 x plus 1")[$y = 6 x + 1$].

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 6 x plus 1")[$y = 6 x + 1$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [], [],
  [1], [], [],
  [#math.equation(block: false, alt: "−2")[$−2$]], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals 6 x plus 1")[$y = 6 x + 1$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [1], [#math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$]],
  [1], [7], [#math.equation(block: false, alt: "open parenthesis 1 , 7 close parenthesis")[$( 1 , 7 )$]],
  [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "−11")[$−11$]], [#math.equation(block: false, alt: "open parenthesis −2 , −11 close parenthesis")[$( −2 , −11 )$]],
))
]
]

#examplebox("Example 7")[][
Complete  to find three solutions to the equation #math.equation(block: false, alt: "5 x minus 4 y equals 20")[$5 x − 4 y = 20$].

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "5 x minus 4 y equals 20")[$5 x − 4 y = 20$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [], [],
  [], [0], [],
  [], [5], [],
))

#solutionbox[
Substitute the given value into the equation #math.equation(block: false, alt: "5 x minus 4 y equals 20")[$5 x − 4 y = 20$] and solve for the other variable. Then, fill in the values in the table. #linebreak() 
#figure(figph[The graph of the equation 5x minus 4y equals 20 is shown for three cases. When x equals 0, y equals negative 5, and the point is (0, negative 5). When y equals 0, x equals 4, and the point is (4, 0). When y equals 5, x equals 8, and the point is (8, 5).], alt: "The graph of the equation 5x minus 4y equals 20 is shown for three cases. When x equals 0, y equals negative 5, and the point is (0, negative 5). When y equals 0, x equals 4, and the point is (4, 0). When y equals 5, x equals 8, and the point is (8, 5).", caption: none)

The results are summarized in .

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "5 x minus 4 y equals 20")[$5 x − 4 y = 20$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$]],
  [4], [0], [#math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]],
  [8], [5], [#math.equation(block: false, alt: "open parenthesis 8 , 5 close parenthesis")[$( 8 , 5 )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Complete the table to find three solutions to this equation: #math.equation(block: false, alt: "2 x minus 5 y equals 20")[$2 x − 5 y = 20$].

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "2 x minus 5 y equals 20")[$2 x − 5 y = 20$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [], [],
  [], [0], [],
  [#math.equation(block: false, alt: "−5")[$−5$]], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "2 x minus 5 y equals 20")[$2 x − 5 y = 20$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$]],
  [10], [0], [#math.equation(block: false, alt: "open parenthesis 10 , 0 close parenthesis")[$( 10 , 0 )$]],
  [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −5 , −6 close parenthesis")[$( −5 , −6 )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Complete the table to find three solutions to this equation: #math.equation(block: false, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$].

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [], [],
  [], [0], [],
  [#math.equation(block: false, alt: "−4")[$−4$]], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [#math.equation(block: false, alt: "−3")[$−3$]], [#math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]],
  [4], [0], [#math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]],
  [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −4 , −6 close parenthesis")[$( −4 , −6 )$]],
))
]
]


=== Find Solutions to a Linear Equation

To find a solution to a linear equation, you really can pick #emph[any] number you want to substitute into the equation for #math.equation(block: false, alt: "x")[$x$] or #math.equation(block: false, alt: "y .")[$y .$] But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

When the equation is in #emph[y]-form, with the #emph[y] by itself on one side of the equation, it is usually easier to choose values of #math.equation(block: false, alt: "x")[$x$] and then solve for #math.equation(block: false, alt: "y")[$y$].

#examplebox("Example 8")[][
Find three solutions to the equation #math.equation(block: false, alt: "y equals −3 x plus 2")[$y = −3 x + 2$].

#solutionbox[
We can substitute any value we want for #math.equation(block: false, alt: "x")[$x$] or any value for #math.equation(block: false, alt: "y")[$y$]. Since the equation is in #emph[y]-form, it will be easier to substitute in values of #math.equation(block: false, alt: "x")[$x$]. Let’s pick #math.equation(block: false, alt: "x equals 0")[$x = 0$], #math.equation(block: false, alt: "x equals 1")[$x = 1$], and #math.equation(block: false, alt: "x equals −1")[$x = −1$]. #linebreak()

#figure(table(
  columns: 6, align: left, inset: 6pt,
  [], [#figure(figph[The expression 'X = Q' is displayed on a white background. 'X' is black, and 'Q' is a light blue, stylized circle with a vertical line through its center.], alt: "The expression 'X = Q' is displayed on a white background. 'X' is black, and 'Q' is a light blue, stylized circle with a vertical line through its center.", caption: none)], [#figure(figph[A close-up image displaying the mathematical equation 'X = 1' in a clear, digital font against a white background.], alt: "A close-up image displaying the mathematical equation 'X = 1' in a clear, digital font against a white background.", caption: none)], [#figure(figph[The mathematical equation 'X = -1' is displayed on a white background. The 'X =' portion is in black text, while the '-1' is presented in a blue-green hue.], alt: "The mathematical equation 'X = -1' is displayed on a white background. The 'X =' portion is in black text, while the '-1' is presented in a blue-green hue.", caption: none)], [], [],
  [], [#figure(figph[The image shows the linear equation y = -3x + 2, representing a straight line with a slope of -3 and a y-intercept of 2.], alt: "The image shows the linear equation y = -3x + 2, representing a straight line with a slope of -3 and a y-intercept of 2.", caption: none)], [#figure(figph[A mathematical equation is displayed, 'y = -3x + 2', representing a linear function with a negative slope and a positive y-intercept.], alt: "A mathematical equation is displayed, 'y = -3x + 2', representing a linear function with a negative slope and a positive y-intercept.", caption: none)], [#figure(figph[The image displays a mathematical equation, 'y = -3x + 2', which represents a linear function in slope-intercept form. The equation is rendered in black text on a plain white background.], alt: "The image displays a mathematical equation, 'y = -3x + 2', which represents a linear function in slope-intercept form. The equation is rendered in black text on a plain white background.", caption: none)], [], [],
  [Substitute the value into the equation.], [#figure(figph[A mathematical expression reads y equals -3 multiplied by 0 plus 2. The digit 0 is highlighted in light blue, indicating a variable or a specific value being substituted.], alt: "A mathematical expression reads y equals -3 multiplied by 0 plus 2. The digit 0 is highlighted in light blue, indicating a variable or a specific value being substituted.", caption: none)], [#figure(figph[A mathematical equation is displayed, showing 'y = -3 \* 1 + 2' with the number 1 highlighted in light blue, indicating it might be a variable or a specific value being substituted.], alt: "A mathematical equation is displayed, showing 'y = -3 * 1 + 2' with the number 1 highlighted in light blue, indicating it might be a variable or a specific value being substituted.", caption: none)], [#figure(figph[A mathematical equation is displayed, showing y = -3 multiplied by -1, plus 2. The -1 is highlighted in a light blue color, indicating it's a substituted value or a point of focus.], alt: "A mathematical equation is displayed, showing y = -3 multiplied by -1, plus 2. The -1 is highlighted in a light blue color, indicating it's a substituted value or a point of focus.", caption: none)], [], [],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing y = 0 + 2.], alt: "A mathematical equation is displayed, showing y = 0 + 2.", caption: none)], [#figure(figph[The image shows a mathematical equation, written in a clear, sans-serif font against a white background. The equation is 'y = -3 + 2'.], alt: "The image shows a mathematical equation, written in a clear, sans-serif font against a white background. The equation is 'y = -3 + 2'.", caption: none)], [#figure(figph[A simple mathematical equation is displayed, showing 'y = 3 + 2' in black text against a white background.], alt: "A simple mathematical equation is displayed, showing 'y = 3 + 2' in black text against a white background.", caption: none)], [], [],
  [Simplify.], [#figure(figph[The equation y = 2 is displayed in black text on a plain white background, representing a horizontal line in a Cartesian coordinate system.], alt: "The equation y = 2 is displayed in black text on a plain white background, representing a horizontal line in a Cartesian coordinate system.", caption: none)], [#figure(figph[The image displays the equation y = -1, rendered in gray text against a plain white background. The equation is centrally positioned, indicating a horizontal line in a Cartesian coordinate system.], alt: "The image displays the equation y = -1, rendered in gray text against a plain white background. The equation is centrally positioned, indicating a horizontal line in a Cartesian coordinate system.", caption: none)], [#figure(figph[The equation y = 5 is displayed in a simple, clear, and centered manner on a white background, representing a horizontal line on a coordinate plane.], alt: "The equation y = 5 is displayed in a simple, clear, and centered manner on a white background, representing a horizontal line on a coordinate plane.", caption: none)], [], [],
  [Write the ordered pair.], [(0, 2)], [(1, −1)], [(−1, 5)], [], [],
  [Check.], [], [], [], [], [],
  [#math.equation(block: false, alt: "y equals −3 x plus 2")[$#h(0.03em) y = −3 x + 2$]], [#math.equation(block: false, alt: "y equals −3 x plus 2")[$#h(0.7em) y = −3 x + 2$]], [#math.equation(block: false, alt: "y equals −3 x plus 2")[$#h(0.04em) y = −3 x + 2$]], [], [], [],
  [#math.equation(block: false, alt: "2 ≟ −3 times 0 plus 2")[$2 ≟ −3 ⋅ 0 + 2$]], [#math.equation(block: false, alt: "−1 ≟ −3 times 1 plus 2")[$−1 ≟ −3 ⋅ 1 + 2$]], [#math.equation(block: false, alt: "5 ≟ −3 open parenthesis −1 close parenthesis plus 2")[$5 ≟ −3 ( −1 ) + 2$]], [], [], [],
  [#math.equation(block: false, alt: "2 ≟ 0 plus 2")[$2 ≟ 0 + 2$]], [#math.equation(block: false, alt: "−1 ≟ −3 plus 2")[$−1 ≟ −3 + 2$]], [#math.equation(block: false, alt: "5 ≟ 3 plus 2")[$5 ≟ 3 + 2$]], [], [], [],
  [#math.equation(block: false, alt: "2 equals 2 ✓")[$2 = 2 ✓$]], [#math.equation(block: false, alt: "−1 equals −1 ✓")[$−1 = −1 ✓$]], [#math.equation(block: false, alt: "5 equals 5 ✓")[$5 = 5 ✓$]], [], [], [],
))

So, #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$], #math.equation(block: false, alt: "open parenthesis 1 , −1 close parenthesis")[$( 1 , −1 )$] and #math.equation(block: false, alt: "open parenthesis −1 , 5 close parenthesis")[$( −1 , 5 )$] are all solutions to #math.equation(block: false, alt: "y equals −3 x plus 2")[$y = −3 x + 2$]. We show them in . #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "y equals −3 x plus 2")[$y = −3 x + 2$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [2], [#math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]],
  [1], [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "open parenthesis 1 , −1 close parenthesis")[$( 1 , −1 )$]],
  [#math.equation(block: false, alt: "−1")[$−1$]], [5], [#math.equation(block: false, alt: "open parenthesis −1 , 5 close parenthesis")[$( −1 , 5 )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three solutions to this equation: #math.equation(block: false, alt: "y equals −2 x plus 3")[$y = −2 x + 3$].

#solutionbox[
Answers will vary.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three solutions to this equation: #math.equation(block: false, alt: "y equals −4 x plus 1")[$y = −4 x + 1$].

#solutionbox[
Answers will vary.
]
]

We have seen how using zero as one value of #math.equation(block: false, alt: "x")[$x$] makes finding the value of #math.equation(block: false, alt: "y")[$y$] easy. When an equation is in standard form, with both the #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] on the same side of the equation, it is usually easier to first find one solution when #math.equation(block: false, alt: "x equals 0")[$x = 0$] find a second solution when #math.equation(block: false, alt: "y equals 0")[$y = 0$], and then find a third solution.

#examplebox("Example 9")[][
Find three solutions to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$].

#solutionbox[
We can substitute any value we want for #math.equation(block: false, alt: "x")[$x$] or any value for #math.equation(block: false, alt: "y")[$y$]. Since the equation is in standard form, let’s pick first #math.equation(block: false, alt: "x equals 0")[$x = 0$], then #math.equation(block: false, alt: "y equals 0")[$y = 0$], and then find a third point. #linebreak()

#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([], [#figure(figph[The equation x = 0 is displayed on a white background.], alt: "The equation x = 0 is displayed on a white background.", caption: none)], [#figure(figph[The equation y = 0 is displayed on a white background.], alt: "The equation y = 0 is displayed on a white background.", caption: none)], [#figure(figph[The image shows the equation 'X = 1' rendered in a dark gray font for the X and equals sign, and a light blue font for the number 1, all against a plain white background.], alt: "The image shows the equation 'X = 1' rendered in a dark gray font for the X and equals sign, and a light blue font for the number 1, all against a plain white background.", caption: none)]),
  [], [#figure(figph[A mathematical equation displays '3x + 2y = 6' in a clear, digital font against a plain white background.], alt: "A mathematical equation displays '3x + 2y = 6' in a clear, digital font against a plain white background.", caption: none)], [#figure(figph[A mathematical equation, 3x + 2y = 6, is displayed against a white background.], alt: "A mathematical equation, 3x + 2y = 6, is displayed against a white background.", caption: none)], [#figure(figph[The image displays the linear equation 3x + 2y = 6 in black text on a white background.], alt: "The image displays the linear equation 3x + 2y = 6 in black text on a white background.", caption: none)],
  [Substitute the value into the equation.], [#figure(figph[An algebraic equation showing 3 multiplied by 0, plus 2y, equals 6. The 0 is highlighted in blue, indicating a substitution or a specific value being considered in the equation.], alt: "An algebraic equation showing 3 multiplied by 0, plus 2y, equals 6. The 0 is highlighted in blue, indicating a substitution or a specific value being considered in the equation.", caption: none)], [#figure(figph[A mathematical equation reads 3x + 2(0) = 6. The number zero within the parentheses is highlighted in red, drawing attention to a specific value or step in the algebraic expression.], alt: "A mathematical equation reads 3x + 2(0) = 6. The number zero within the parentheses is highlighted in red, drawing attention to a specific value or step in the algebraic expression.", caption: none)], [#figure(figph[A mathematical equation shows '3(1) + 2y = 6', where 3 multiplied by 1 is added to 2y, equaling 6. The number 1 is highlighted in light blue.], alt: "A mathematical equation shows '3(1) + 2y = 6', where 3 multiplied by 1 is added to 2y, equaling 6. The number 1 is highlighted in light blue.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the mathematical equation '0 + 2y = 6' in black text on a white background.], alt: "The image shows the mathematical equation '0 + 2y = 6' in black text on a white background.", caption: none)], [#figure(figph[The image shows a mathematical equation, '3x + 0 = 6', which simplifies to 3x = 6, and solving for x gives x = 2. The equation is rendered in black text against a white background.], alt: "The image shows a mathematical equation, '3x + 0 = 6', which simplifies to 3x = 6, and solving for x gives x = 2. The equation is rendered in black text against a white background.", caption: none)], [#figure(figph[A mathematical equation is displayed with the expression '3 + 2y = 6'. The equation consists of the numbers 3, 2, and 6, the variable y, and the operators for addition and equality.], alt: "A mathematical equation is displayed with the expression '3 + 2y = 6'. The equation consists of the numbers 3, 2, and 6, the variable y, and the operators for addition and equality.", caption: none)],
  [Solve.], [#figure(figph[The image displays the algebraic equation '2y = 6' on a white background, demonstrating a simple linear equation where twice a variable 'y' equals six.], alt: "The image displays the algebraic equation '2y = 6' on a white background, demonstrating a simple linear equation where twice a variable 'y' equals six.", caption: none)], [#figure(figph[The mathematical equation '3x = 6' is displayed on a white background.], alt: "The mathematical equation '3x = 6' is displayed on a white background.", caption: none)], [#figure(figph[The image displays a simple algebraic equation, 2y = 3, in black text on a white background.], alt: "The image displays a simple algebraic equation, 2y = 3, in black text on a white background.", caption: none)],
  [], [#figure(figph[The image displays the mathematical equation 'y = 3' in a simple, clear font on a plain white background.], alt: "The image displays the mathematical equation 'y = 3' in a simple, clear font on a plain white background.", caption: none)], [#figure(figph[The equation 'x = 2' is shown in black text against a white background.], alt: "The equation 'x = 2' is shown in black text against a white background.", caption: none)], [#figure(figph[The image displays the mathematical equation y = 3/2 on a white background.], alt: "The image displays the mathematical equation y = 3/2 on a white background.", caption: none)],
  [Write the ordered pair.], [(0, 3)], [(2, 0)], [#math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 1 , frac(3, 2) )$]],
))

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Check.], [], [],
  [#math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6 #h(1.3em)$]], [#math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6 #h(1.3em)$]], [#math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6 #h(1.3em)$]],
  [#math.equation(block: false, alt: "3 times 0 plus 2 times 3 ≟ 6")[$3 ⋅ 0 + 2 ⋅ 3 ≟ 6 #h(1.3em)$]], [#math.equation(block: false, alt: "3 times 2 plus 2 times 0 ≟ 6")[$3 ⋅ 2 + 2 ⋅ 0 ≟ 6 #h(1.3em)$]], [#math.equation(block: false, alt: "3 times 1 plus 2 times the fraction 3 over 2 ≟ 6")[$3 ⋅ 1 + 2 ⋅ frac(3, 2) ≟ 6 #h(1.3em)$]],
  [#math.equation(block: false, alt: "0 plus 6 ≟ 6")[$0 + 6 ≟ 6 #h(1.3em)$]], [#math.equation(block: false, alt: "6 plus 0 ≟ 6")[$6 + 0 ≟ 6 #h(1.3em)$]], [#math.equation(block: false, alt: "3 plus 3 ≟ 6")[$3 + 3 ≟ 6 #h(1.3em)$]],
  [#math.equation(block: false, alt: "6 equals 6 ✓")[$6 = 6 ✓$]], [#math.equation(block: false, alt: "6 equals 6 ✓")[$6 = 6 ✓$]], [#math.equation(block: false, alt: "6 equals 6 ✓")[$6 = 6 ✓$]],
))

So #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$], #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$], and #math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 1 , frac(3, 2) )$] are all solutions to the equation #math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]. We can list these three solutions in . #linebreak()

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[#math.equation(block: false, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]]], [], [],
  [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]],
  [0], [3], [#math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]],
  [2], [0], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]],
  [1], [#math.equation(block: false, alt: "the fraction 3 over 2")[$frac(3, 2)$]], [#math.equation(block: false, alt: "open parenthesis 1 , the fraction 3 over 2 close parenthesis")[$( 1 , frac(3, 2) )$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three solutions to the equation #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$].

#solutionbox[
Answers will vary.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find three solutions to the equation #math.equation(block: false, alt: "4 x plus 2 y equals 8")[$4 x + 2 y = 8$].

#solutionbox[
Answers will vary.
]
]


=== Key Concepts

- #strong[Sign Patterns of the Quadrants] #linebreak() 
#math.equation(block: false, alt: "Quadrant I, Quadrant II, Quadrant III, Quadrant IV; open parenthesis x , y close parenthesis, open parenthesis x , y close parenthesis, open parenthesis x , y close parenthesis, open parenthesis x , y close parenthesis; open parenthesis plus , plus close parenthesis, open parenthesis − , plus close parenthesis, open parenthesis − , − close parenthesis, open parenthesis plus , − close parenthesis")[$"Quadrant I" &  &  & "Quadrant II" &  &  & "Quadrant III" &  &  & "Quadrant IV" \ ( x , y ) &  &  & ( x , y ) &  &  & ( x , y ) &  &  & ( x , y ) \ ( + , + ) &  &  & ( "−" , + ) &  &  & ( "−" , "−" ) &  &  & ( + , "−" )$]
- #strong[Points on the Axes] #linebreak() 
- On the #emph[x]-axis, #math.equation(block: false, alt: "y equals 0")[$y = 0$]. Points with a #emph[y]-coordinate equal to 0 are on the #emph[x]-axis, and have coordinates #math.equation(block: false, alt: "open parenthesis a , 0 close parenthesis")[$( a , 0 )$].
- On the #emph[y]-axis, #math.equation(block: false, alt: "x equals 0")[$x = 0$]. Points with an #emph[x]-coordinate equal to 0 are on the #emph[y]-axis, and have coordinates #math.equation(block: false, alt: "open parenthesis 0 , b close parenthesis .")[$( 0 , b ) .$]
- #strong[Solution of a Linear Equation] #linebreak() 
- An ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] is a solution of the linear equation #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$], if the equation is a true statement when the #emph[x]- and #emph[y]- values of the ordered pair are substituted into the equation.


==== Practice Makes Perfect

#strong[Plot Points in a Rectangular Coordinate System]

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

+ ⓐ #math.equation(block: false, alt: "open parenthesis −4 , 2 close parenthesis")[$( −4 , 2 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −2 close parenthesis")[$( −1 , −2 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis 3 , −5 close parenthesis")[$( 3 , −5 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis −3 , 5 close parenthesis")[$( −3 , 5 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 3 , 2 close parenthesis")[$( frac(5, 3) , 2 )$]

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 2) is plotted and labeled "a". The point (negative 1, negative 2) is plotted and labeled "b". The point (3, negative 5) is plotted and labeled "c". The point (negative 3, 5) is plotted and labeled “d”. The point (5 thirds, 2) is plotted and labeled “e”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 2) is plotted and labeled \"a\". The point (negative 1, negative 2) is plotted and labeled \"b\". The point (3, negative 5) is plotted and labeled \"c\". The point (negative 3, 5) is plotted and labeled “d”. The point (5 thirds, 2) is plotted and labeled “e”.", caption: none)
]

+ ⓐ #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 3 , −3 close parenthesis")[$( 3 , −3 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 1 close parenthesis")[$( −4 , 1 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis 4 , −1 close parenthesis")[$( 4 , −1 )$]
+ ⓔ #math.equation(block: false, alt: "open parenthesis the fraction 3 over 2 , 1 close parenthesis")[$( frac(3, 2) , 1 )$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 3 , −1 close parenthesis")[$( 3 , −1 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 1 close parenthesis")[$( −3 , 1 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis −2 , 2 close parenthesis")[$( −2 , 2 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis −4 , −3 close parenthesis")[$( −4 , −3 )$]
+ ⓔ #math.equation(block: false, alt: "open parenthesis 1 , the fraction 14 over 5 close parenthesis")[$( 1 , frac(14, 5) )$]

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (3, negative 1) is plotted and labeled "a". The point (negative 3, 1) is plotted and labeled "b". The point (negative 2, 2) is plotted and labeled "c". The point (negative 4, negative 3) is plotted and labeled “d”. The point (1, 14 fifths) is plotted and labeled “e”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (3, negative 1) is plotted and labeled \"a\". The point (negative 3, 1) is plotted and labeled \"b\". The point (negative 2, 2) is plotted and labeled \"c\". The point (negative 4, negative 3) is plotted and labeled “d”. The point (1, 14 fifths) is plotted and labeled “e”.", caption: none)
]

+ ⓐ #math.equation(block: false, alt: "open parenthesis −1 , 1 close parenthesis")[$( −1 , 1 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis 1 , −4 close parenthesis")[$( 1 , −4 )$]
+ ⓔ #math.equation(block: false, alt: "open parenthesis 3 , the fraction 7 over 2 close parenthesis")[$( 3 , frac(7, 2) )$]

In the following exercises, plot each point in a rectangular coordinate system.

+ ⓐ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$]
+ ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 2, 0) is plotted and labeled "a". The point (negative 3, 0) is plotted and labeled "b". The point (0, 0) is plotted and labeled "c". The point (0, 4) is plotted and labeled “d”. The point (0, 3) is plotted and labeled “e”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 2, 0) is plotted and labeled \"a\". The point (negative 3, 0) is plotted and labeled \"b\". The point (0, 0) is plotted and labeled \"c\". The point (0, 4) is plotted and labeled “d”. The point (0, 3) is plotted and labeled “e”.", caption: none)
]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis −1 , 0 close parenthesis")[$( −1 , 0 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]
+ ⓔ #math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 0 close parenthesis")[$( −4 , 0 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis 1 , 0 close parenthesis")[$( 1 , 0 )$]
+ ⓔ #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]

#solutionbox[
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, 0) is plotted and labeled "a". The point (0, negative 3) is plotted and labeled "b". The point (negative 4, 0) is plotted and labeled "c". The point (1, 0) is plotted and labeled “d”. The point (0, negative 2) is plotted and labeled “e”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, 0) is plotted and labeled \"a\". The point (0, negative 3) is plotted and labeled \"b\". The point (negative 4, 0) is plotted and labeled \"c\". The point (1, 0) is plotted and labeled “d”. The point (0, negative 2) is plotted and labeled “e”.", caption: none)
]

+ ⓐ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]
+ ⓓ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]
+ ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled “A”. The point (negative 3, negative 4) is plotted and labeled “B”. The point (1, negative 3) is plotted and labeled “C”. The point (4, 3) is plotted and labeled “D”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (negative 4, 1) is plotted and labeled “A”. The point (negative 3, negative 4) is plotted and labeled “B”. The point (1, negative 3) is plotted and labeled “C”. The point (4, 3) is plotted and labeled “D”.", caption: none)

#solutionbox[
A: #math.equation(block: false, alt: "open parenthesis −4 , 1 close parenthesis")[$( −4 , 1 )$] B: #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] C: #math.equation(block: false, alt: "open parenthesis 1 , −3 close parenthesis")[$( 1 , −3 )$] D: #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$]
]

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The point (negative 4, 2) is plotted and labeled “A”. The point (3, 5) is plotted and labeled “B”. The point (negative 3, negative 2) is plotted and labeled “C”. The point (5, negative 1) is plotted and labeled “D”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The point (negative 4, 2) is plotted and labeled “A”. The point (3, 5) is plotted and labeled “B”. The point (negative 3, negative 2) is plotted and labeled “C”. The point (5, negative 1) is plotted and labeled “D”.", caption: none)

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 2) is plotted and labeled “A”. The point (negative 2, 0) is plotted and labeled “B”. The point (0, 5) is plotted and labeled “C”. The point (5, 0) is plotted and labeled “D”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 2) is plotted and labeled “A”. The point (negative 2, 0) is plotted and labeled “B”. The point (0, 5) is plotted and labeled “C”. The point (5, 0) is plotted and labeled “D”.", caption: none)

#solutionbox[
A: #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$] B: #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] C: #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] D: #math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$]
]

#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 1) is plotted and labeled “A”. The point (negative 1, 0) is plotted and labeled “B”. The point (4, 0) is plotted and labeled “C”. The point (0, 4) is plotted and labeled “D”.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (0, negative 1) is plotted and labeled “A”. The point (negative 1, 0) is plotted and labeled “B”. The point (4, 0) is plotted and labeled “C”. The point (0, 4) is plotted and labeled “D”.", caption: none)

#strong[Verify Solutions to an Equation in Two Variables]

In the following exercises, which ordered pairs are solutions to the given equations?

#math.equation(block: true, alt: "2 x plus y equals 6")[$2 x + y = 6$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$]

#solutionbox[
a, b
]

#math.equation(block: true, alt: "x plus 3 y equals 9")[$x + 3 y = 9$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis −3 , −3 close parenthesis")[$( −3 , −3 )$]

#math.equation(block: true, alt: "4 x minus 2 y equals 8")[$4 x − 2 y = 8$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$]

#solutionbox[
a, c
]

#math.equation(block: true, alt: "3 x minus 2 y equals 12")[$3 x − 2 y = 12$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$]

#math.equation(block: true, alt: "y equals 4 x plus 3")[$y = 4 x + 3$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −1 close parenthesis")[$( −1 , −1 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 2 , 5 close parenthesis")[$( frac(1, 2) , 5 )$]

#solutionbox[
b, c
]

#math.equation(block: true, alt: "y equals 2 x minus 5")[$y = 2 x − 5$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 2 , −4 close parenthesis")[$( frac(1, 2) , −4 )$]

#math.equation(block: true, alt: "y equals the fraction 1 over 2 x minus 1")[$y = frac(1, 2) x − 1$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis −6 , −4 close parenthesis")[$( −6 , −4 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis −4 , −1 close parenthesis")[$( −4 , −1 )$]

#solutionbox[
a, b
]

#math.equation(block: true, alt: "y equals the fraction 1 over 3 x plus 1")[$y = frac(1, 3) x + 1$]

+ ⓐ #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$]
+ ⓑ #math.equation(block: false, alt: "open parenthesis 9 , 4 close parenthesis")[$( 9 , 4 )$]
+ ⓒ #math.equation(block: false, alt: "open parenthesis −6 , −1 close parenthesis")[$( −6 , −1 )$]

#strong[Complete a Table of Solutions to a Linear Equation]

In the following exercises, complete the table to find solutions to each linear equation.

#math.equation(block: true, alt: "y equals 2 x minus 4")[$y = 2 x − 4$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [2], [], [],
  [#math.equation(block: false, alt: "−1")[$−1$]], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [#math.equation(block: false, alt: "−4")[$−4$]], [#math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$]],
  [2], [0], [#math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$]],
  [#math.equation(block: false, alt: "−1")[$−1$]], [#math.equation(block: false, alt: "−6")[$−6$]], [#math.equation(block: false, alt: "open parenthesis −1 , −6 close parenthesis")[$( −1 , −6 )$]],
))
]

#math.equation(block: true, alt: "y equals 3 x minus 1")[$y = 3 x − 1$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [2], [], [],
  [#math.equation(block: false, alt: "−1")[$−1$]], [], [],
))

#math.equation(block: true, alt: "y equals − x plus 5")[$y = "−" x + 5$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [3], [], [],
  [#math.equation(block: false, alt: "−2")[$−2$]], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [5], [#math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]],
  [3], [2], [#math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$]],
  [#math.equation(block: false, alt: "−2")[$−2$]], [7], [#math.equation(block: false, alt: "open parenthesis −2 , 7 close parenthesis")[$( −2 , 7 )$]],
))
]

#math.equation(block: true, alt: "y equals − x plus 2")[$y = "−" x + 2$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [3], [], [],
  [#math.equation(block: false, alt: "−2")[$−2$]], [], [],
))

#math.equation(block: true, alt: "y equals the fraction 1 over 3 x plus 1")[$y = frac(1, 3) x + 1$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [3], [], [],
  [6], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [1], [#math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$]],
  [3], [2], [#math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$]],
  [6], [3], [#math.equation(block: false, alt: "open parenthesis 6 , 3 close parenthesis")[$( 6 , 3 )$]],
))
]

#math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus 4")[$y = frac(1, 2) x + 4$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [2], [], [],
  [4], [], [],
))

#math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x minus 2")[$y = − frac(3, 2) x − 2$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [2], [], [],
  [#math.equation(block: false, alt: "−2")[$−2$]], [], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]],
  [2], [#math.equation(block: false, alt: "−5")[$−5$]], [#math.equation(block: false, alt: "open parenthesis 2 , −5 close parenthesis")[$( 2 , −5 )$]],
  [#math.equation(block: false, alt: "−2")[$−2$]], [1], [#math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$]],
))
]

#math.equation(block: true, alt: "y equals minus the fraction 2 over 3 x minus 1")[$y = − frac(2, 3) x − 1$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [3], [], [],
  [#math.equation(block: false, alt: "−3")[$−3$]], [], [],
))

#math.equation(block: true, alt: "x plus 3 y equals 6")[$x + 3 y = 6$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [3], [], [],
  [], [0], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [2], [#math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]],
  [3], [4], [#math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis")[$( 3 , 1 )$]],
  [6], [0], [#math.equation(block: false, alt: "open parenthesis 6 , 0 close parenthesis")[$( 6 , 0 )$]],
))
]

#math.equation(block: true, alt: "x plus 2 y equals 8")[$x + 2 y = 8$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [4], [], [],
  [], [0], [],
))

#math.equation(block: true, alt: "2 x minus 5 y equals 10")[$2 x − 5 y = 10$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [10], [], [],
  [], [0], [],
))

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [#math.equation(block: false, alt: "−2")[$−2$]], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]],
  [10], [2], [#math.equation(block: false, alt: "open parenthesis 10 , 2 close parenthesis")[$( 10 , 2 )$]],
  [5], [0], [#math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$]],
))
]

#math.equation(block: true, alt: "3 x minus 4 y equals 12")[$3 x − 4 y = 12$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [], [],
  [8], [], [],
  [], [0], [],
))

#strong[Find Solutions to a Linear Equation]

In the following exercises, find three solutions to each linear equation.

#math.equation(block: true, alt: "y equals 5 x minus 8")[$y = 5 x − 8$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "y equals 3 x minus 9")[$y = 3 x − 9$]

#math.equation(block: true, alt: "y equals −4 x plus 5")[$y = −4 x + 5$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "y equals −2 x plus 7")[$y = −2 x + 7$]

#math.equation(block: true, alt: "x plus y equals 8")[$x + y = 8$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "x plus y equals 6")[$x + y = 6$]

#math.equation(block: true, alt: "x plus y equals −2")[$x + y = −2$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "x plus y equals −1")[$x + y = −1$]

#math.equation(block: true, alt: "3 x plus y equals 5")[$3 x + y = 5$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "2 x plus y equals 3")[$2 x + y = 3$]

#math.equation(block: true, alt: "4 x minus y equals 8")[$4 x − y = 8$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "5 x minus y equals 10")[$5 x − y = 10$]

#math.equation(block: true, alt: "2 x plus 4 y equals 8")[$2 x + 4 y = 8$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "3 x plus 2 y equals 6")[$3 x + 2 y = 6$]

#math.equation(block: true, alt: "5 x minus 2 y equals 10")[$5 x − 2 y = 10$]

#solutionbox[
Answers will vary.
]

#math.equation(block: true, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$]


==== Everyday Math

#strong[Weight of a baby.] Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

ⓐ Plot the points on a coordinate plane.

#figure(figph[A blank Cartesian coordinate plane featuring a grid. The x-axis is labeled from 0 to 12, and the y-axis is labeled from 0 to 25, ready for data plotting.], alt: "A blank Cartesian coordinate plane featuring a grid. The x-axis is labeled from 0 to 12, and the y-axis is labeled from 0 to 25, ready for data plotting.", caption: none)

ⓑ Why is only Quadrant I needed?

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[Age #math.equation(block: false, alt: "x")[$x$]]], [#strong[Weight #math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [0], [7], [(0, 7)],
  [2], [11], [(2, 11)],
  [4], [15], [(4, 15)],
  [6], [16], [(6, 16)],
  [8], [19], [(8, 19)],
  [10], [20], [(10, 20)],
  [12], [21], [(12, 21)],
))

#solutionbox[
ⓐ  #linebreak() 
#figure(figph[The graph shows the x y-coordinate plane. The x- and y-axes each run from 0 to 25. The points (0, 7), (2, 11), (4, 15), (6, 16), (8, 19), (10, 20) and (12, 21) are plotted and labeled.], alt: "The graph shows the x y-coordinate plane. The x- and y-axes each run from 0 to 25. The points (0, 7), (2, 11), (4, 15), (6, 16), (8, 19), (10, 20) and (12, 21) are plotted and labeled.", caption: none)

 #linebreak() 
ⓑ Age and weight are only positive.
]

#strong[Weight of a child.] Latresha recorded her son’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

ⓐ Plot the points on a coordinate plane.

#figure(figph[An empty 5x5 grid on a Cartesian coordinate plane with x and y axes marked from 0 to 50, showing increments of 10.], alt: "An empty 5x5 grid on a Cartesian coordinate plane with x and y axes marked from 0 to 50, showing increments of 10.", caption: none)

ⓑ Why is only Quadrant I needed?

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([#strong[Height #math.equation(block: false, alt: "x")[$x$]]], [#strong[Weight #math.equation(block: false, alt: "y")[$y$]]], [#strong[#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]]]),
  [28], [22], [(28, 22)],
  [31], [27], [(31, 27)],
  [33], [33], [(33, 33)],
  [37], [35], [(37, 35)],
  [40], [41], [(40, 41)],
  [42], [45], [(42, 45)],
))


==== Writing Exercises

Explain in words how you plot the point #math.equation(block: false, alt: "open parenthesis 4 , −2 close parenthesis")[$( 4 , −2 )$] in a rectangular coordinate system.

#solutionbox[
Answers will vary.
]

How do you determine if an ordered pair is a solution to a given equation?

Is the point #math.equation(block: false, alt: "open parenthesis −3 , 0 close parenthesis")[$( −3 , 0 )$] on the #emph[x]-axis or #emph[y]-axis? How do you know?

#solutionbox[
Answers will vary.
]

Is the point #math.equation(block: false, alt: "open parenthesis 0 , 8 close parenthesis")[$( 0 , 8 )$] on the #emph[x]-axis or #emph[y]-axis? How do you know?


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “plot points in a rectangular coordinate system,”, “identify points on a graph,” “verify solutions to an equation in two variables,” “complete a table of solutions to a linear equation,” and “find solutions to a linear equation.” The rest of the cells are blank.], alt: "This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “plot points in a rectangular coordinate system,”, “identify points on a graph,” “verify solutions to an equation in two variables,” “complete a table of solutions to a linear equation,” and “find solutions to a linear equation.” The rest of the cells are blank.", caption: none)

ⓑ If most of your checks were:

#strong[…confidently.] Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

#strong[…with some help.] This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

#strong[…no, I don’t get it.] This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.