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== 3.3#h(0.6em)Find the Equation of a Line

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

Simplify: #math.equation(block: false, alt: "the fraction 2 over 5 open parenthesis x plus 15 close parenthesis .")[$frac(2, 5) ( x + 15 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 5 x plus 6")[$frac(2, 5) x + 6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "−3 open parenthesis x minus open parenthesis −2 close parenthesis close parenthesis .")[$−3 ( x − ( −2 ) ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 3 x minus 6")[$− 3 x − 6$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve for #emph[y]: #math.equation(block: false, alt: "y minus 3 equals −2 open parenthesis x plus 1 close parenthesis .")[$y − 3 = −2 ( x + 1 ) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "y equals minus 2 x plus 1")[$y = − 2 x + 1$]
]
]

How do online companies know that “you may also like” a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.

The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section, we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.


=== Find an Equation of the Line Given the Slope and #emph[y]-Intercept

We can easily determine the slope and intercept of a line if the equation is written in slope-intercept form, #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$] Now we will do the reverse—we will start with the slope and #emph[y]-intercept and use them to find the equation of the line.

#examplebox("Example 1")[][
Find the equation of a line with slope #math.equation(block: false, alt: "−9")[$−9$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis .")[$( 0 , −4 ) .$]

#solutionbox[
Since we are given the slope and #emph[y]-intercept of the line, we can substitute the needed values into the slope-intercept form, #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Name the slope.], [#figure(figph[The equation 'm = -9' is displayed on a white background. The letter 'm' and the equals sign are in gray, while the number '-9' is prominently shown in red.], alt: "The equation 'm = -9' is displayed on a white background. The letter 'm' and the equals sign are in gray, while the number '-9' is prominently shown in red.", caption: none)]),
  [Name the #emph[y]-intercept.], [#figure(figph[The text], alt: "The text", caption: none)],
  [Substitute the values into #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$]], [#figure(figph[The equation for a straight line, y = mx + b, is shown with 'm' (slope) in red and 'b' (y-intercept) in blue.], alt: "The equation for a straight line, y = mx + b, is shown with 'm' (slope) in red and 'b' (y-intercept) in blue.", caption: none)],
  [], [#figure(figph[A mathematical equation is displayed, reading y equals negative 9x plus parentheses negative 4 parentheses. The -9 is in red and the -4 is in light blue.], alt: "A mathematical equation is displayed, reading y equals negative 9x plus parentheses negative 4 parentheses. The -9 is in red and the -4 is in light blue.", caption: none)],
  [], [#figure(figph[The image displays a linear equation in slope-intercept form, y = -9x - 4, which represents a straight line with a slope of -9 and a y-intercept of -4.], alt: "The image displays a linear equation in slope-intercept form, y = -9x - 4, which represents a straight line with a slope of -9 and a y-intercept of -4.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line with slope #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis .")[$( 0 , 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 2 over 5 x plus 4")[$y = frac(2, 5) x + 4$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line with slope #math.equation(block: false, alt: "−1")[$−1$] and #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis .")[$( 0 , −3 ) .$]

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

Sometimes, the slope and intercept need to be determined from the graph.

#examplebox("Example 2")[][
Find the equation of the line shown in the graph.

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 6), (0, negative 4), (3, negative 2), and (6, 0).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 3, negative 6), (0, negative 4), (3, negative 2), and (6, 0).", caption: none)

#solutionbox[
We need to find the slope and #emph[y]-intercept of the line from the graph so we can substitute the needed values into the slope-intercept form, #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$]

To find the slope, we choose two points on the graph.

The #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] and the graph passes through #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis .")[$( 3 , −2 ) .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the slope, by counting the rise and run.], [#figure(figph[The formula for calculating the slope (m) of a line, represented as the ratio of its vertical change (rise) to its horizontal change (run): m = rise / run.], alt: "The formula for calculating the slope (m) of a line, represented as the ratio of its vertical change (rise) to its horizontal change (run): m = rise / run.", caption: none)]),
  [], [#figure(figph[The equation m equals the fraction 2 over 3 is displayed on a white background.], alt: "The equation m equals the fraction 2 over 3 is displayed on a white background.", caption: none)],
  [Find the #emph[y]-intercept.], [#figure(figph[The text 'y-intercept (0, -4)' is displayed in the image, indicating the point where a line or curve crosses the y-axis.], alt: "The text 'y-intercept (0, -4)' is displayed in the image, indicating the point where a line or curve crosses the y-axis.", caption: none)],
  [Substitute the values into #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$]], [#figure(figph[The image shows the mathematical equation for a straight line in slope-intercept form, which is 'y = mx + b'. The variable 'm' is colored red, and 'b' is colored blue.], alt: "The image shows the mathematical equation for a straight line in slope-intercept form, which is 'y = mx + b'. The variable 'm' is colored red, and 'b' is colored blue.", caption: none)],
  [], [#figure(figph[The image shows the linear equation y = (2/3)x - 4. The slope (2/3) is colored red, and the y-intercept (-4) is colored light blue, highlighting key components of the equation.], alt: "The image shows the linear equation y = (2/3)x - 4. The slope (2/3) is colored red, and the y-intercept (-4) is colored light blue, highlighting key components of the equation.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of the line shown in the graph.

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 5, negative 2), (0, 1), and (5, 4).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 5, negative 2), (0, 1), and (5, 4).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 3 over 5 x plus 1")[$y = frac(3, 5) x + 1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of the line shown in the graph.

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 5), (3, negative 1), and (6, 3).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 5), (3, negative 1), and (6, 3).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 4 over 3 x minus 5")[$y = frac(4, 3) x − 5$]
]
]


=== Find an Equation of the Line Given the Slope and a Point

Finding an equation of a line using the slope-intercept form of the equation works well when you are given the slope and #emph[y]-intercept or when you read them off a graph. But what happens when you have another point instead of the #emph[y]-intercept?

We are going to use the slope formula to derive another form of an equation of the line.

Suppose we have a line that has slope #emph[m] and that contains some specific point #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] and some other point, which we will just call #math.equation(block: false, alt: "open parenthesis x , y close parenthesis .")[$( x , y ) .$] We can write the slope of this line and then change it to a different form.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "m equals the fraction y minus y sub 1 over x minus x sub 1")[$#h(3.2em) m = frac(y − y_(1), x − x_(1))$]]),
  [Multiply both sides of the equation by #math.equation(block: false, alt: "x minus x sub 1 .")[$x − x_(1) .$]], [#math.equation(block: false, alt: "m open parenthesis x minus x sub 1 close parenthesis equals open parenthesis the fraction y minus y sub 1 over x minus x sub 1 close parenthesis open parenthesis x minus x sub 1 close parenthesis")[$m ( x − x_(1) ) = ( frac(y − y_(1), x − x_(1)) ) ( x − x_(1) )$]],
  [Simplify.], [#math.equation(block: false, alt: "m open parenthesis x minus x sub 1 close parenthesis equals y minus y sub 1")[$m ( x − x_(1) ) = y − y_(1)$]],
  [Rewrite the equation with the #math.equation(block: false, alt: "y")[$y$] terms on the left.], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$#h(1.4em) y − y_(1) = m ( x − x_(1) )$]],
))

This format is called the #strong[point-slope form] of an equation of a line.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Point-slope Form of an Equation of a Line]

The #strong[point-slope form] of an equation of a line with slope #emph[m] and containing the point #math.equation(block: false, alt: "open parenthesis x sub 1 , y sub 1 close parenthesis")[$( x_(1) , y_(1) )$] is:

#math.equation(block: true, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]
]

We can use the point-slope form of an equation to find an equation of a line when we know the slope and at least one point. Then, we will rewrite the equation in slope-intercept form. Most applications of linear equations use the the slope-intercept form.

#examplebox("Example 3")[How to Find an Equation of a Line Given a Point and the Slope][
Find an equation of a line with slope #math.equation(block: false, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$] that contains the point #math.equation(block: false, alt: "open parenthesis 6 , −4 close parenthesis .")[$( 6 , −4 ) .$] Write the equation in slope-intercept form.

#solutionbox[
#figure(figph[Step 1 is to identify the slope. The slope is given. m equals negative 1 divided by 3.], alt: "Step 1 is to identify the slope. The slope is given. m equals negative 1 divided by 3.", caption: none)

#figure(figph[Step 2 is to identify the point. The point is given. x 1 is 6 and y 1 is negative 4.], alt: "Step 2 is to identify the point. The point is given. x 1 is 6 and y 1 is negative 4.", caption: none)

#figure(figph[Step 3 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus negative 4 equals negative 1 divided by 3 times the quantity x minus 6 in parentheses. This simplifies to y plus 4 equals negative 1 divided by 3 times x plus 2.], alt: "Step 3 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus negative 4 equals negative 1 divided by 3 times the quantity x minus 6 in parentheses. This simplifies to y plus 4 equals negative 1 divided by 3 times x plus 2.", caption: none)

#figure(figph[Step 4 is to write the equation in slope-intercept form. y equals negative 1 divided by 3 times x minus 2.], alt: "Step 4 is to write the equation in slope-intercept form. y equals negative 1 divided by 3 times x minus 2.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line with slope #math.equation(block: false, alt: "m equals minus the fraction 2 over 5")[$m = − frac(2, 5)$] and containing the point #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis .")[$( 10 , −5 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x minus 1")[$y = − frac(2, 5) x − 1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line with slope #math.equation(block: false, alt: "m equals minus the fraction 3 over 4 ,")[$m = − frac(3, 4) ,$] and containing the point #math.equation(block: false, alt: "open parenthesis 4 , −7 close parenthesis .")[$( 4 , −7 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 3 over 4 x minus 4")[$y = − frac(3, 4) x − 4$]
]
]

We list the steps for easy reference.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To find an equation of a line given the slope and a point.]

+ Identify the slope.
+ Identify the point.
+ Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]
+ Write the equation in slope-intercept form.
]

#examplebox("Example 4")[][
Find an equation of a horizontal line that contains the point #math.equation(block: false, alt: "open parenthesis −2 , −6 close parenthesis .")[$( −2 , −6 ) .$] Write the equation in slope-intercept form.

#solutionbox[
Every horizontal line has slope 0. We can substitute the slope and points into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the slope.], [#figure(figph[The mathematical equation 'm = ø' is displayed in black and light blue text on a plain white background, symbolizing the variable 'm' equaling the empty set or null value.], alt: "The mathematical equation 'm = ø' is displayed in black and light blue text on a plain white background, symbolizing the variable 'm' equaling the empty set or null value.", caption: none)]),
  [Identify the point.], [#figure(figph[Coordinate pair (-2, -6), displayed as x₁ = -2 and y₁ = -6.], alt: "Coordinate pair (-2, -6), displayed as x₁ = -2 and y₁ = -6.", caption: none)],
  [Substitute the values into #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]], [#figure(figph[The point-slope form of a linear equation: y - y1 = m(x - x1), where m represents the slope and (x1, y1) is a known point on the line.], alt: "The point-slope form of a linear equation: y - y1 = m(x - x1), where m represents the slope and (x1, y1) is a known point on the line.", caption: none)],
  [], [#figure(figph[A mathematical equation is displayed: y - (-6) = 0(x - (-2)). The numbers -6 and -2 are highlighted in red, and the 0 is highlighted in light blue.], alt: "A mathematical equation is displayed: y - (-6) = 0(x - (-2)). The numbers -6 and -2 are highlighted in red, and the 0 is highlighted in light blue.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the equation y + 6 = 0 against a plain white background.], alt: "The image displays the equation y + 6 = 0 against a plain white background.", caption: none)],
  [], [#figure(figph[The image displays the algebraic equation y = -6 in a simple, clear font against a white background.], alt: "The image displays the algebraic equation y = -6 in a simple, clear font against a white background.", caption: none)],
  [Write in slope-intercept form.], [It is in #emph[y]-form, but could be written #math.equation(block: false, alt: "y equals 0 x minus 6 .")[$y = 0 x − 6 .$]],
))

Did we end up with the form of a horizontal line, #math.equation(block: false, alt: "y equals b ?")[$y = b ?$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a horizontal line containing the point #math.equation(block: false, alt: "open parenthesis −3 , 8 close parenthesis .")[$( −3 , 8 ) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a horizontal line containing the point #math.equation(block: false, alt: "open parenthesis −1 , 4 close parenthesis .")[$( −1 , 4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y equals 4")[$y = 4$]
]
]


=== Find an Equation of the Line Given Two Points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.

So far, we have two options for finding an equation of a line: slope-intercept or point-slope. When we start with two points, it makes more sense to use the point-slope form.

But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.

#examplebox("Example 5")[How to Find the Equation of a Line Given Two Points][
Find an equation of a line that contains the points #math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$] and #math.equation(block: false, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$] Write the equation in slope-intercept form.

#solutionbox[
#figure(figph[Step 1 is to find the slope using the given points. Find the slope of the line through (negative 3, negative 1) and (2, and negative 2). m equals the quotient of y 2 minus y 1 in parentheses and x 2 minus x 1 in parentheses. m equals the quotient of negative 2 minus negative 1 in parentheses and 2 minus negative 3 in parentheses. m equals negative 1 divided by 5.], alt: "Step 1 is to find the slope using the given points. Find the slope of the line through (negative 3, negative 1) and (2, and negative 2). m equals the quotient of y 2 minus y 1 in parentheses and x 2 minus x 1 in parentheses. m equals the quotient of negative 2 minus negative 1 in parentheses and 2 minus negative 3 in parentheses. m equals negative 1 divided by 5.", caption: none)

#figure(figph[Step 2 is to identify the point. Choose either point. x 1 is 2 and y 1 is negative 2.], alt: "Step 2 is to identify the point. Choose either point. x 1 is 2 and y 1 is negative 2.", caption: none)

#figure(figph[Step 3 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus negative 2 equals negative 1 divided by 5 times the quantity x minus 2 in parentheses. This simplifies to y plus 2 equals negative 1 divided by 5 times x plus 2 divided by 5.], alt: "Step 3 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus negative 2 equals negative 1 divided by 5 times the quantity x minus 2 in parentheses. This simplifies to y plus 2 equals negative 1 divided by 5 times x plus 2 divided by 5.", caption: none)

#figure(figph[Step 4 is to write the equation in slope-intercept form. y equals negative 1 divided by 5 times x minus 8 divided by 5.], alt: "Step 4 is to write the equation in slope-intercept form. y equals negative 1 divided by 5 times x minus 8 divided by 5.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line containing the points #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 1 , −3 close parenthesis .")[$( 1 , −3 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 1 over 3 x minus the fraction 10 over 3")[$y = frac(1, 3) x − frac(10, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line containing the points #math.equation(block: false, alt: "open parenthesis −4 , −3 close parenthesis")[$( −4 , −3 )$] and #math.equation(block: false, alt: "open parenthesis 1 , −5 close parenthesis .")[$( 1 , −5 ) .$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x minus the fraction 23 over 5")[$y = − frac(2, 5) x − frac(23, 5)$]
]
]

The steps are summarized here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To find an equation of a line given two points.]

+ Find the slope using the given points. #math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]
+ Choose one point.
+ Substitute the values into the point-slope form: #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]
+ Write the equation in slope-intercept form.
]

#examplebox("Example 6")[][
Find an equation of a line that contains the points #math.equation(block: false, alt: "open parenthesis −3 , 5 close parenthesis")[$( −3 , 5 )$] and #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis .")[$( −3 , 4 ) .$] Write the equation in slope-intercept form.

#solutionbox[
Again, the first step will be to find the slope.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the slope of the line through #math.equation(block: false, alt: "open parenthesis −3 , 5 close parenthesis")[$( −3 , 5 )$] and #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis .")[$( −3 , 4 ) .$]], [#math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]]),
  [], [#math.equation(block: false, alt: "m equals the fraction 4 minus 5 over −3 minus open parenthesis −3 close parenthesis")[$m = frac(4 − 5, −3 − ( −3 ))$]],
  [], [#math.equation(block: false, alt: "m equals the fraction −1 over 0")[$m = frac(−1, 0)$]],
  [], [The slope is undefined.],
))

This tells us it is a vertical line. Both of our points have an #emph[x]-coordinate of #math.equation(block: false, alt: "minus 3 .")[$− 3 .$] So our equation of the line is #math.equation(block: false, alt: "x equals −3 .")[$x = −3 .$] Since there is no #emph[y], we cannot write it in slope-intercept form.

You may want to sketch a graph using the two given points. Does your graph agree with our conclusion that this is a vertical line?
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line containing the points #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 5 , −4 close parenthesis .")[$( 5 , −4 ) .$]

#solutionbox[
#math.equation(block: true, alt: "x equals 5")[$x = 5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the equation of a line containing the points #math.equation(block: false, alt: "open parenthesis −4 , 4 close parenthesis")[$( −4 , 4 )$] and #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis .")[$( −4 , 3 ) .$]

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

We have seen that we can use either the slope-intercept form or the point-slope form to find an equation of a line. Which form we use will depend on the information we are given.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [To Write an Equation of a Line], [], [],
  [#strong[If given:]], [#strong[Use:]], [#strong[Form:]],
  [Slope and #emph[y]-intercept], [slope-intercept], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]],
  [Slope and a point], [point-slope], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]],
  [Two points], [point-slope], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]],
))


=== Find an Equation of a Line Parallel to a Given Line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point-slope equation.

First, let’s look at this graphically.

This graph shows #math.equation(block: false, alt: "y equals 2 x minus 3 .")[$y = 2 x − 3 .$] We want to graph a line parallel to this line and passing through the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis .")[$( −2 , 1 ) .$]

#figure(figph[This figure has a graph of a straight line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The point (negative 2, 1) is plotted. The line does not go through the point (negative 2, 1).], alt: "This figure has a graph of a straight line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The point (negative 2, 1) is plotted. The line does not go through the point (negative 2, 1).", caption: none)

We know that parallel lines have the same slope. So the second line will have the same slope as #math.equation(block: false, alt: "y equals 2 x minus 3 .")[$y = 2 x − 3 .$] That slope is #math.equation(block: false, alt: "m sub ∥ equals 2 .")[$m_(∥) = 2 .$] We’ll use the notation #math.equation(block: false, alt: "m sub ∥")[$m_(∥)$] to represent the slope of a line parallel to a line with slope #emph[m]. (Notice that the subscript || looks like two parallel lines.)

The second line will pass through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and have #math.equation(block: false, alt: "m equals 2 .")[$m = 2 .$]

To graph the line, we start at#math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and count out the rise and run.

With #math.equation(block: false, alt: "m equals 2")[$m = 2$] (or #math.equation(block: false, alt: "m equals the fraction 2 over 1")[$m = frac(2, 1)$]), we count out the rise 2 and the run 1. We draw the line, as shown in the graph.

#figure(figph[This figure has a graph of a two straight lines on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The first line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The points (negative 2, 1) and (negative 1, 3) are plotted. The second line goes through the points (negative 2, 1) and (negative 1, 3).], alt: "This figure has a graph of a two straight lines on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The first line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The points (negative 2, 1) and (negative 1, 3) are plotted. The second line goes through the points (negative 2, 1) and (negative 1, 3).", caption: none)

Do the lines appear parallel? Does the second line pass through#math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis ?")[$( −2 , 1 ) ?$]

We were asked to graph the line, now let’s see how to do this algebraically.

We can use either the slope-intercept form or the point-slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point-slope form.

#examplebox("Example 7")[How to Find the Equation of a Line Parallel to a Given Line and a Point][
Find an equation of a line parallel to #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis .")[$( −2 , 1 ) .$] Write the equation in slope-intercept form.

#solutionbox[
#figure(figph[Step 1 is to find the slope of the given line. The line is in slope-intercept form, y equals 2 x minus 3. m equals 2.], alt: "Step 1 is to find the slope of the given line. The line is in slope-intercept form, y equals 2 x minus 3. m equals 2.", caption: none)

#figure(figph[Step 2 is to find the slope of the parallel line. Parallel lines have the same slope. m equals 2.], alt: "Step 2 is to find the slope of the parallel line. Parallel lines have the same slope. m equals 2.", caption: none)

#figure(figph[Step 3 is to identify the point. The given point is (negative 2, 1). x 1 is negative 2 and y 1 is 1.], alt: "Step 3 is to identify the point. The given point is (negative 2, 1). x 1 is negative 2 and y 1 is 1.", caption: none)

#figure(figph[Step 4 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus 1 equals 2 times the quantity x minus negative 2 in parentheses. This simplifies to y minus 1 equals 2 x plus 4.], alt: "Step 4 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus 1 equals 2 times the quantity x minus negative 2 in parentheses. This simplifies to y minus 1 equals 2 x plus 4.", caption: none)

#figure(figph[Step 5 is to write the equation in slope-intercept form. y equals 2 x plus 5.], alt: "Step 5 is to write the equation in slope-intercept form. y equals 2 x plus 5.", caption: none)

Look at graph with the parallel lines shown previously. Does this equation make sense? What is the #emph[y]-intercept of the line? What is the slope?
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line parallel to the line #math.equation(block: false, alt: "y equals 3 x plus 1")[$y = 3 x + 1$] that contains the point #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis .")[$( 4 , 2 ) .$] Write the equation in slope-intercept form.

#solutionbox[
#math.equation(block: true, alt: "y equals 3 x minus 10")[$y = 3 x − 10$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line parallel to the line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x minus 3")[$y = frac(1, 2) x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis 6 , 4 close parenthesis .")[$( 6 , 4 ) .$]

Write the equation in slope-intercept form.

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Find an equation of a line parallel to a given line.]

+ Find the slope of the given line.
+ Find the slope of the parallel line.
+ Identify the point.
+ Substitute the values into the point-slope form: #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]
+ Write the equation in slope-intercept form.
]


=== Find an Equation of a Line Perpendicular to a Given Line

Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point-slope equation, like we did with parallel lines.

This graph shows #math.equation(block: false, alt: "y equals 2 x minus 3 .")[$y = 2 x − 3 .$] Now, we want to graph a line perpendicular to this line and passing through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis .")[$( −2 , 1 ) .$]

#figure(figph[This figure has a graph of a straight line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The point (negative 2, 1) is plotted. The line does not go through the point (negative 2, 1).], alt: "This figure has a graph of a straight line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The point (negative 2, 1) is plotted. The line does not go through the point (negative 2, 1).", caption: none)

We know that perpendicular lines have slopes that are negative reciprocals.

We’ll use the notation #math.equation(block: false, alt: "m sub ⊥")[$m_(⊥)$] to represent the slope of a line perpendicular to a line with slope #emph[m]. (Notice that the subscript #math.equation(block: false, alt: "⊥")[$⊥$] looks like the right angles made by two perpendicular lines.)

#math.equation(block: true, alt: "y equals 2 x minus 3, perpendicular line; m equals 2, m sub ⊥ equals minus the fraction 1 over 2")[$y = 2 x − 3 &  &  & "perpendicular line" \ m = 2 &  &  & m_(⊥) = − frac(1, 2)$]We now know the perpendicular line will pass through #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] with #math.equation(block: false, alt: "m sub ⊥ equals minus the fraction 1 over 2 .")[$m_(⊥) = − frac(1, 2) .$]

To graph the line, we will start at #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and count out the rise #math.equation(block: false, alt: "−1")[$−1$] and the run 2. Then we draw the line.

#figure(figph[This figure has a graph of two perpendicular straight lines on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The first line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The points (negative 2, 1) and (0, 0) are plotted. A right triangle is drawn connecting the points (negative 2, 1), (negative 2, 0), and (0, 0). The second line goes through the points (negative 2, 1) and (0, 0).], alt: "This figure has a graph of two perpendicular straight lines on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The first line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The points (negative 2, 1) and (0, 0) are plotted. A right triangle is drawn connecting the points (negative 2, 1), (negative 2, 0), and (0, 0). The second line goes through the points (negative 2, 1) and (0, 0).", caption: none)

Do the lines appear perpendicular? Does the second line pass through#math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis ?")[$( −2 , 1 ) ?$]

We were asked to graph the line, now, let’s see how to do this algebraically.

We can use either the slope-intercept form or the point-slope form to find an equation of a line. In this example we know one point, and can find the slope, so we will use the point-slope form.

#examplebox("Example 8")[How to Find the Equation of a Line Perpendicular to a Given Line and a Point][
Find an equation of a line perpendicular to #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis .")[$( −2 , 1 ) .$] Write the equation in slope-intercept form.

#solutionbox[
#figure(figph[Step 1 is to find the slope of the given line. The line is in slope-intercept form, y equals 2 x minus 3. m equals 2.], alt: "Step 1 is to find the slope of the given line. The line is in slope-intercept form, y equals 2 x minus 3. m equals 2.", caption: none)

#figure(figph[Step 2 is to find the slope of the perpendicular line. The slopes of perpendicular lines are negative reciprocals. m equals negative 1 divided by 2], alt: "Step 2 is to find the slope of the perpendicular line. The slopes of perpendicular lines are negative reciprocals. m equals negative 1 divided by 2", caption: none)

#figure(figph[Step 3 is to identify the point. The given point is (negative 2, 1). x 1 is negative 2 and y 1 is 1.], alt: "Step 3 is to identify the point. The given point is (negative 2, 1). x 1 is negative 2 and y 1 is 1.", caption: none)

#figure(figph[Step 4 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus 1 equals negative 1 divided by 2 times the quantity x minus negative 2 in parentheses. This simplifies to y minus 1 equals negative 1 divided by 2 times the quantity x plus 2 in parentheses. This further simplifies to y minus 1 equals negative 1 divided by 2 times x minus 1.], alt: "Step 4 is to substitute the values into the point-slope form y minus y 1 equals m times the quantity x minus x 1 in parentheses. y minus 1 equals negative 1 divided by 2 times the quantity x minus negative 2 in parentheses. This simplifies to y minus 1 equals negative 1 divided by 2 times the quantity x plus 2 in parentheses. This further simplifies to y minus 1 equals negative 1 divided by 2 times x minus 1.", caption: none)

#figure(figph[Step 5 is to write the equation in slope-intercept form. y equals negative 1 divided by 2 times x.], alt: "Step 5 is to write the equation in slope-intercept form. y equals negative 1 divided by 2 times x.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line perpendicular to the line #math.equation(block: false, alt: "y equals 3 x plus 1")[$y = 3 x + 1$] that contains the point #math.equation(block: false, alt: "open parenthesis 4 , 2 close parenthesis .")[$( 4 , 2 ) .$] Write the equation in slope-intercept form.

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 1 over 3 x plus the fraction 10 over 3")[$y = − frac(1, 3) x + frac(10, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line perpendicular to the line #math.equation(block: false, alt: "y equals the fraction 1 over 2 x minus 3")[$y = frac(1, 2) x − 3$] that contains the point #math.equation(block: false, alt: "open parenthesis 6 , 4 close parenthesis .")[$( 6 , 4 ) .$] Write the equation in slope-intercept form.

#solutionbox[
#math.equation(block: true, alt: "y equals −2 x plus 16")[$y = −2 x + 16$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Find an equation of a line perpendicular to a given line.]

+ Find the slope of the given line.
+ Find the slope of the perpendicular line.
+ Identify the point.
+ Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]
+ Write the equation in slope-intercept form.
]

#examplebox("Example 9")[][
Find an equation of a line perpendicular to #math.equation(block: false, alt: "x equals 5")[$x = 5$] that contains the point #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis .")[$( 3 , −2 ) .$] Write the equation in slope-intercept form.

#solutionbox[
Again, since we know one point, the point-slope option seems more promising than the slope-intercept option. We need the slope to use this form, and we know the new line will be perpendicular to #math.equation(block: false, alt: "x equals 5 .")[$x = 5 .$] This line is vertical, so its perpendicular will be horizontal. This tells us the #math.equation(block: false, alt: "m sub ⊥ equals 0 .")[$m_(⊥) = 0 .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Identify the point.], [#math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis")[$#h(1.5em) ( 3 , −2 )$]]),
  [Identify the slope of the perpendicular line.], [#math.equation(block: false, alt: "m sub ⊥ equals 0")[$#h(0.5em) m_(⊥) = 0$]],
  [Substitute the values into #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$#h(3.3em) y − y_(1) = m ( x − x_(1) )$]],
  [], [#math.equation(block: false, alt: "y minus open parenthesis −2 close parenthesis equals 0 open parenthesis x minus 3 close parenthesis")[$#h(1.5em) y − ( −2 ) = 0 ( x − 3 )$]],
  [Simplify.], [#math.equation(block: false, alt: "y plus 2 equals 0")[$y + 2 = 0$]],
  [], [#math.equation(block: false, alt: "y equals −2")[$#h(2.5em) y = −2$]],
))

Sketch the graph of both lines. On your graph, do the lines appear to be perpendicular?
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "x equals 4")[$x = 4$] that contains the point #math.equation(block: false, alt: "open parenthesis 4 , −5 close parenthesis .")[$( 4 , −5 ) .$]. Write the equation in slope-intercept form.

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "x equals 2")[$x = 2$] that contains the point #math.equation(block: false, alt: "open parenthesis 2 , −1 close parenthesis .")[$( 2 , −1 ) .$] Write the equation in slope-intercept form.

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

In , we used the point-slope form to find the equation. We could have looked at this in a different way.

We want to find a line that is perpendicular to #math.equation(block: false, alt: "x equals 5")[$x = 5$] that contains the point #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis .")[$( 3 , −2 ) .$] This graph shows us the line#math.equation(block: false, alt: "x equals 5")[$x = 5$] and the point #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis .")[$( 3 , −2 ) .$]

#figure(figph[This figure has a graph of a straight vertical line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (5, 0), (5, 1), and (5, 2). The point (3, negative 2) is plotted. The line does not go through the point (3, negative 2).], alt: "This figure has a graph of a straight vertical line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (5, 0), (5, 1), and (5, 2). The point (3, negative 2) is plotted. The line does not go through the point (3, negative 2).", caption: none)

We know every line perpendicular to a vertical line is horizontal, so we will sketch the horizontal line through #math.equation(block: false, alt: "open parenthesis 3 , −2 close parenthesis .")[$( 3 , −2 ) .$]

#figure(figph[This figure has a graph of a straight vertical line and a straight horizontal line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The vertical line goes through the points (5, 0), (5, 1), and (5, 2). The horizontal line goes through the points (negative 2, negative 2), (0, negative 2), (3, negative 2), and (6, negative 2).], alt: "This figure has a graph of a straight vertical line and a straight horizontal line on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The vertical line goes through the points (5, 0), (5, 1), and (5, 2). The horizontal line goes through the points (negative 2, negative 2), (0, negative 2), (3, negative 2), and (6, negative 2).", caption: none)

Do the lines appear perpendicular?

If we look at a few points on this horizontal line, we notice they all have #emph[y]-coordinates of #math.equation(block: false, alt: "−2 .")[$−2 .$] So, the equation of the line perpendicular to the vertical line #math.equation(block: false, alt: "x equals 5")[$x = 5$] is #math.equation(block: false, alt: "y equals −2 .")[$y = −2 .$]

#examplebox("Example 10")[][
Find an equation of a line that is perpendicular to #math.equation(block: false, alt: "y equals −3")[$y = −3$] that contains the point #math.equation(block: false, alt: "open parenthesis −3 , 5 close parenthesis .")[$( −3 , 5 ) .$] Write the equation in slope-intercept form.

#solutionbox[
The line #math.equation(block: false, alt: "y equals −3")[$y = −3$] is a horizontal line. Any line perpendicular to it must be vertical, in the form #math.equation(block: false, alt: "x equals a .")[$x = a .$] Since the perpendicular line is vertical and passes through #math.equation(block: false, alt: "open parenthesis −3 , 5 close parenthesis ,")[$( −3 , 5 ) ,$] every point on it has an #emph[x]-coordinate of #math.equation(block: false, alt: "−3 .")[$−3 .$] The equation of the perpendicular line is #math.equation(block: false, alt: "x equals −3")[$x = −3$]

You may want to sketch the lines. Do they appear perpendicular?
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "y equals 1")[$y = 1$] that contains the point #math.equation(block: false, alt: "open parenthesis −5 , 1 close parenthesis .")[$( −5 , 1 ) .$] Write the equation in slope-intercept form.

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find an equation of a line that is perpendicular to the line #math.equation(block: false, alt: "y equals −5")[$y = −5$] that contains the point #math.equation(block: false, alt: "open parenthesis −4 , −5 close parenthesis .")[$( −4 , −5 ) .$] Write the equation in slope-intercept form.

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with finding the equation of a line.

- #link("https://openstax.org/l/37slopeycept")[Write an Equation of Line Given its slope and Y-Intercept]
- #link("https://openstax.org/l/37twoptspline")[Find the equation given two points]
- #link("https://openstax.org/l/37perpenpara")[Find the equation of perpendicular and parallel lines]
]


=== Key Concepts

- #strong[How to find an equation of a line given the slope and a point.]
+ Identify the slope.
+ Identify the point.
+ Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]
+ Write the equation in slope-intercept form. #linebreak()  #linebreak()  #linebreak()  #linebreak()
- #strong[How to find an equation of a line given two points.]
+ Find the slope using the given points. #math.equation(block: false, alt: "m equals the fraction y sub 2 minus y sub 1 over x sub 2 minus x sub 1")[$m = frac(y_(2) − y_(1), x_(2) − x_(1))$]
+ Choose one point.
+ Substitute the values into the point-slope form: #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]
+ Write the equation in slope-intercept form. #linebreak() 
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [To Write an Equation of a Line], [], [],
  [#strong[If given:]], [#strong[Use:]], [#strong[Form:]],
  [Slope and #emph[y]-intercept], [#strong[slope-intercept]], [#math.equation(block: false, alt: "y equals m x plus b")[$y = m x + b$]],
  [Slope and a point], [#strong[point-slope]], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]],
  [Two points], [#strong[point-slope]], [#math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]],
))
- #strong[How to find an equation of a line parallel to a given line.]
+ Find the slope of the given line.
+ Find the slope of the parallel line.
+ Identify the point.
+ Substitute the values into the point-slope form: #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis .")[$y − y_(1) = m ( x − x_(1) ) .$]
+ Write the equation in slope-intercept form
- #strong[How to find an equation of a line perpendicular to a given line.]
+ Find the slope of the given line.
+ Find the slope of the perpendicular line.
+ Identify the point.
+ Substitute the values into the point-slope form, #math.equation(block: false, alt: "y minus y sub 1 equals m open parenthesis x minus x sub 1 close parenthesis")[$y − y_(1) = m ( x − x_(1) )$]
+ Write the equation in slope-intercept form.


==== Practice Makes Perfect

#strong[Find an Equation of the Line Given the Slope and #emph[y]-Intercept]

In the following exercises, find the equation of a line with given slope and #emph[y]-intercept. Write the equation in slope-intercept form.

slope 3 and #linebreak() #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]

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

slope 8 and #linebreak() #emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −6 close parenthesis")[$( 0 , −6 )$]

slope #math.equation(block: false, alt: "−3")[$−3$] and #linebreak() #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]

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

slope #math.equation(block: false, alt: "−1")[$−1$] and #linebreak() #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]

slope #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$] and #linebreak() #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −5 close parenthesis")[$( 0 , −5 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 1 over 5 x minus 5")[$y = frac(1, 5) x − 5$]
]

slope #math.equation(block: false, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] and #linebreak() #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$]

slope 0 and #linebreak() #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −1 close parenthesis")[$( 0 , −1 )$]

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

slope #math.equation(block: false, alt: "−4")[$−4$] and #linebreak() #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope-intercept form.

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 5), (1, negative 2), and (2, 1).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 5), (1, negative 2), and (2, 1).", caption: none)

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

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 4), (1, 2), and (2, 0).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 4), (1, 2), and (2, 0).", caption: none)

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (2, negative 2), and (6, 0).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (2, negative 2), and (6, 0).", caption: none)

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

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (4, 5), and (8, 8).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (4, 5), and (8, 8).", caption: none)

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 3), (3, negative 1), and (6, negative 5).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 3), (3, negative 1), and (6, negative 5).", caption: none)

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 4 over 3 x plus 3")[$y = − frac(4, 3) x + 3$]
]

#figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 1), (2, negative 4), and (4, negative 7).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 1), (2, negative 4), and (4, negative 7).", caption: none)

#figure(figph[This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).], alt: "This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).", caption: none)

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

#figure(figph[This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 6), (1, 6), and (2, 6).], alt: "This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 6), (1, 6), and (2, 6).", caption: none)

#strong[Find an Equation of the Line Given the Slope and a Point]

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form.

#math.equation(block: false, alt: "m equals the fraction 5 over 8 ,")[$m = frac(5, 8) ,$] point #math.equation(block: false, alt: "open parenthesis 8 , 3 close parenthesis")[$( 8 , 3 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 5 over 8 x minus 2")[$y = frac(5, 8) x − 2$]
]

#math.equation(block: false, alt: "m equals the fraction 5 over 6 ,")[$m = frac(5, 6) ,$] point #math.equation(block: false, alt: "open parenthesis 6 , 7 close parenthesis")[$( 6 , 7 )$]

#math.equation(block: false, alt: "m equals minus the fraction 3 over 5 ,")[$m = − frac(3, 5) ,$] point #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis")[$( 10 , −5 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 3 over 5 x plus 1")[$y = − frac(3, 5) x + 1$]
]

#math.equation(block: false, alt: "m equals minus the fraction 3 over 4 ,")[$m = − frac(3, 4) ,$] point #math.equation(block: false, alt: "open parenthesis 8 , −5 close parenthesis")[$( 8 , −5 )$]

#math.equation(block: false, alt: "m equals minus the fraction 3 over 2 ,")[$m = − frac(3, 2) ,$] point #math.equation(block: false, alt: "open parenthesis −4 , −3 close parenthesis")[$( −4 , −3 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x minus 9")[$y = − frac(3, 2) x − 9$]
]

#math.equation(block: false, alt: "m equals minus the fraction 5 over 2 ,")[$m = − frac(5, 2) ,$] point #math.equation(block: false, alt: "open parenthesis −8 , −2 close parenthesis")[$( −8 , −2 )$]

#math.equation(block: false, alt: "m equals −7 ,")[$m = −7 ,$] point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals −7 x minus 10")[$y = −7 x − 10$]
]

#math.equation(block: false, alt: "m equals −4 ,")[$m = −4 ,$] point #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$]

Horizontal line containing #math.equation(block: false, alt: "open parenthesis −2 , 5 close parenthesis")[$( −2 , 5 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals 5")[$y = 5$]
]

Horizontal line containing #math.equation(block: false, alt: "open parenthesis −2 , −3 close parenthesis")[$( −2 , −3 )$]

Horizontal line containing #math.equation(block: false, alt: "open parenthesis −1 , −7 close parenthesis")[$( −1 , −7 )$]

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

Horizontal line containing #math.equation(block: false, alt: "open parenthesis 4 , −8 close parenthesis")[$( 4 , −8 )$]

#strong[Find an Equation of the Line Given Two Points]

In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.

#math.equation(block: false, alt: "open parenthesis 2 , 6 close parenthesis")[$( 2 , 6 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 3 close parenthesis")[$( 5 , 3 )$]

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

#math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 8 , 1 close parenthesis")[$( 8 , 1 )$]

#math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 5 , minus 2 close parenthesis")[$( 5 , − 2 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 1 over 4 x minus the fraction 13 over 4")[$y = frac(1, 4) x − frac(13, 4)$]
]

#math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] and #math.equation(block: false, alt: "open parenthesis 4 , −6 close parenthesis")[$( 4 , −6 )$]

#math.equation(block: false, alt: "open parenthesis −1 , 3 close parenthesis")[$( −1 , 3 )$] and #math.equation(block: false, alt: "open parenthesis −6 , −7 close parenthesis")[$( −6 , −7 )$]

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

#math.equation(block: false, alt: "open parenthesis −2 , 8 close parenthesis")[$( −2 , 8 )$] and #math.equation(block: false, alt: "open parenthesis −4 , −6 close parenthesis")[$( −4 , −6 )$]

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

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 7 over 2 x plus 4")[$y = − frac(7, 2) x + 4$]
]

#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$] and #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$]

#math.equation(block: false, alt: "open parenthesis 7 , 2 close parenthesis")[$( 7 , 2 )$] and #math.equation(block: false, alt: "open parenthesis 7 , −2 close parenthesis")[$( 7 , −2 )$]

#solutionbox[
#math.equation(block: true, alt: "x equals 7")[$x = 7$]
]

#math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis")[$( −2 , 1 )$] and #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$]

#math.equation(block: false, alt: "open parenthesis 3 , −4 close parenthesis")[$( 3 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 5 , −4 close parenthesis")[$( 5 , −4 )$]

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

#math.equation(block: false, alt: "open parenthesis −6 , −3 close parenthesis")[$( −6 , −3 )$] and #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$]

#strong[Find an Equation of a Line Parallel to a Given Line]

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form.

line #math.equation(block: false, alt: "y equals 4 x plus 2 ,")[$y = 4 x + 2 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$]

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

line #math.equation(block: false, alt: "y equals −3 x minus 1 ,")[$y = −3 x − 1 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis .")[$( 2 , −3 ) .$]

line #math.equation(block: false, alt: "2 x minus y equals 6 ,")[$2 x − y = 6 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis .")[$( 3 , 0 ) .$]

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

line #math.equation(block: false, alt: "2 x plus 3 y equals 6 ,")[$2 x + 3 y = 6 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis .")[$( 0 , 5 ) .$]

line #math.equation(block: false, alt: "x equals −4 ,")[$x = −4 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis −3 , −5 close parenthesis .")[$( −3 , −5 ) .$]

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

line #math.equation(block: false, alt: "x minus 2 equals 0 ,")[$x − 2 = 0 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 1 , −2 close parenthesis")[$( 1 , −2 )$]

line #math.equation(block: false, alt: "y equals 5 ,")[$y = 5 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 2 , −2 close parenthesis")[$( 2 , −2 )$]

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

line #math.equation(block: false, alt: "y plus 2 equals 0 ,")[$y + 2 = 0 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 3 , −3 close parenthesis")[$( 3 , −3 )$]

#strong[Find an Equation of a Line Perpendicular to a Given Line]

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form.

line #math.equation(block: false, alt: "y equals −2 x plus 3 ,")[$y = −2 x + 3 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 2 , 2 close parenthesis")[$( 2 , 2 )$]

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

line #math.equation(block: false, alt: "y equals − x plus 5 ,")[$y = "−" x + 5 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$]

line #math.equation(block: false, alt: "y equals the fraction 3 over 4 x minus 2 ,")[$y = frac(3, 4) x − 2 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 4 over 3 x")[$y = − frac(4, 3) x$]
]

line #math.equation(block: false, alt: "y equals the fraction 2 over 3 x minus 4 ,")[$y = frac(2, 3) x − 4 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 2 , −4 close parenthesis")[$( 2 , −4 )$]

line #math.equation(block: false, alt: "2 x minus 3 y equals 8 ,")[$2 x − 3 y = 8 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 4 , −1 close parenthesis")[$( 4 , −1 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x plus 5")[$y = − frac(3, 2) x + 5$]
]

line #math.equation(block: false, alt: "4 x minus 3 y equals 5 ,")[$4 x − 3 y = 5 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis −3 , 2 close parenthesis")[$( −3 , 2 )$]

line #math.equation(block: false, alt: "2 x plus 5 y equals 6 ,")[$2 x + 5 y = 6 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 5 over 2 x")[$y = frac(5, 2) x$]
]

line #math.equation(block: false, alt: "4 x plus 5 y equals −3 ,")[$4 x + 5 y = −3 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]

line #math.equation(block: false, alt: "x equals 3 ,")[$x = 3 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 3 , 4 close parenthesis")[$( 3 , 4 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals 4")[$y = 4$]
]

line #math.equation(block: false, alt: "x equals −5 ,")[$x = −5 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis 1 , −2 close parenthesis")[$( 1 , −2 )$]

line #math.equation(block: false, alt: "x equals 7 ,")[$x = 7 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$]

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

line #math.equation(block: false, alt: "x equals −1 ,")[$x = −1 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis −4 , 0 close parenthesis")[$( −4 , 0 )$]

line #math.equation(block: false, alt: "y minus 3 equals 0 ,")[$y − 3 = 0 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$]

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

line #math.equation(block: false, alt: "y minus 6 equals 0 ,")[$y − 6 = 0 ,$] #linebreak() point #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$]

line #emph[y]-axis, #linebreak() point #math.equation(block: false, alt: "open parenthesis 3 , 4 close parenthesis")[$( 3 , 4 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals 4")[$y = 4$]
]

line #emph[y]-axis, #linebreak() point #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$]

#strong[Mixed Practice]

In the following exercises, find the equation of each line. Write the equation in slope-intercept form.

Containing the points #math.equation(block: false, alt: "open parenthesis 4 , 3 close parenthesis")[$( 4 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 8 , 1 close parenthesis")[$( 8 , 1 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 1 over 2 x plus 5")[$y = − frac(1, 2) x + 5$]
]

Containing the points #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −3 , −2 close parenthesis")[$( −3 , −2 )$]

#math.equation(block: false, alt: "m equals the fraction 1 over 6 ,")[$m = frac(1, 6) ,$] containing point #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 1 over 6 x")[$y = frac(1, 6) x$]
]

#math.equation(block: false, alt: "m equals the fraction 5 over 6 ,")[$m = frac(5, 6) ,$] containing point #math.equation(block: false, alt: "open parenthesis 6 , 7 close parenthesis")[$( 6 , 7 )$]

Parallel to the line #math.equation(block: false, alt: "4 x plus 3 y equals 6 ,")[$4 x + 3 y = 6 ,$] containing point #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 4 over 3 x minus 3")[$y = − frac(4, 3) x − 3$]
]

Parallel to the line #math.equation(block: false, alt: "2 x plus 3 y equals 6 ,")[$2 x + 3 y = 6 ,$] containing point #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]

#math.equation(block: false, alt: "m equals minus the fraction 3 over 4 ,")[$m = − frac(3, 4) ,$] containing point #math.equation(block: false, alt: "open parenthesis 8 , −5 close parenthesis")[$( 8 , −5 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 3 over 4 x plus 1")[$y = − frac(3, 4) x + 1$]
]

#math.equation(block: false, alt: "m equals minus the fraction 3 over 5 ,")[$m = − frac(3, 5) ,$] containing point #math.equation(block: false, alt: "open parenthesis 10 , −5 close parenthesis")[$( 10 , −5 )$]

Perpendicular to the line #math.equation(block: false, alt: "y minus 1 equals 0 ,")[$y − 1 = 0 ,$] point #math.equation(block: false, alt: "open parenthesis −2 , 6 close parenthesis")[$( −2 , 6 )$]

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

Perpendicular to the line #emph[y]-axis, point #math.equation(block: false, alt: "open parenthesis −6 , 2 close parenthesis")[$( −6 , 2 )$]

Parallel to the line #math.equation(block: false, alt: "x equals −3 ,")[$x = −3 ,$] containing point #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$]

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

Parallel to the line #math.equation(block: false, alt: "x equals −4 ,")[$x = −4 ,$] containing point #math.equation(block: false, alt: "open parenthesis −3 , −5 close parenthesis")[$( −3 , −5 )$]

Containing the points #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis")[$( −3 , −4 )$] and #math.equation(block: false, alt: "open parenthesis 2 , −5 close parenthesis")[$( 2 , −5 )$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 1 over 5 x minus the fraction 23 over 5")[$y = − frac(1, 5) x − frac(23, 5)$]
]

Containing the points #math.equation(block: false, alt: "open parenthesis −5 , −3 close parenthesis")[$( −5 , −3 )$] and #math.equation(block: false, alt: "open parenthesis 4 , −6 close parenthesis")[$( 4 , −6 )$]

Perpendicular to the line #math.equation(block: false, alt: "x minus 2 y equals 5 ,")[$x − 2 y = 5 ,$] point #math.equation(block: false, alt: "open parenthesis −2 , 2 close parenthesis")[$( −2 , 2 )$]

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

Perpendicular to the line #math.equation(block: false, alt: "4 x plus 3 y equals 1 ,")[$4 x + 3 y = 1 ,$] point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]


==== Writing Exercises

Why are all horizontal lines parallel?

#solutionbox[
Answers will vary.
]

Explain in your own words why the slopes of two perpendicular lines must have opposite signs.


==== Self Check

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

#figure(figph[The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “confidently”, the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “find the equation of the line given the slope and y-intercept”, “find an equation of the line given the slope and a point”, “find an equation of the line given two points”, “find an equation of a line parallel to a given line”, and “find an equation of a line perpendicular to a given line”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.], alt: "The figure shows a table with six rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “confidently”, the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “find the equation of the line given the slope and y-intercept”, “find an equation of the line given the slope and a point”, “find an equation of the line given two points”, “find an equation of a line parallel to a given line”, and “find an equation of a line perpendicular to a given line”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?