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== 2.3#h(0.6em)Solving Equations Using the Subtraction and Addition Properties of Equality

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

#math.equation(block: false, alt: "Evaluate x plus 8 when x equals 11 .")[$"Evaluate" #h(0.2em) x + 8 #h(0.2em) "when" #h(0.2em) x = 11 .$] #linebreak() If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "19")[$19$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "Evaluate 5 x minus 3 when x equals 9 .")[$"Evaluate" #h(0.2em) 5 x − 3 #h(0.2em) "when" #h(0.2em) x = 9 .$] #linebreak() If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "42")[$42$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into algebra: the difference of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "8 .")[$8 .$] #linebreak() If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x minus 8")[$x − 8$]
]
]

When some people hear the word #emph[algebra], they think of solving equations. The applications of solving equations are limitless and extend to all careers and fields. In this section, we will begin solving equations. We will start by solving basic equations, and then as we proceed through the course we will build up our skills to cover many different forms of equations.


=== Determine Whether a Number is a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a #strong[solution] of the equation. It is the answer to the puzzle!

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

A #strong[solution to an equation] is a value of a variable that makes a true statement when substituted into the equation.

The process of finding the solution to an equation is called solving the equation.
]

To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of #math.equation(block: false, alt: "x plus 2 equals 7 ?")[$x + 2 = 7 ?$] If you said #math.equation(block: false, alt: "5 ,")[$5 ,$] you’re right! We say #math.equation(block: false, alt: "5")[$5$] is a solution to the equation #math.equation(block: false, alt: "x plus 2 equals 7")[$x + 2 = 7$] because when we substitute #math.equation(block: false, alt: "5")[$5$] for #math.equation(block: false, alt: "x")[$x$] the resulting statement is true.

#math.equation(block: true, alt: "x plus 2 equals 7; 5 plus 2 equals ? 7; 7 equals 7 ✓")[$x + 2 = 7 \ 5 + 2 limits(=)^(?) 7 \  \ #h(3em) 7 = 7 ✓$]Since #math.equation(block: false, alt: "5 plus 2 equals 7")[$5 + 2 = 7$] is a true statement, we know that #math.equation(block: false, alt: "5")[$5$] is indeed a solution to the equation.

The symbol #math.equation(block: false, alt: "equals ?")[$limits(=)^(?)$] asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign #math.equation(block: false, alt: "(=)")[$"(=)"$] or not-equal sign #math.equation(block: false, alt: "(≠).")[$"(≠)."$]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Determine whether a number is a solution to an equation.]

+ Substitute the number for the variable in the equation.
+ Simplify the expressions on both sides of the equation.
+ Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
]

#examplebox("Example 1")[][
#math.equation(block: true, alt: "Determine whether x equals 5 is a solution of 6 x minus 17 equals 16 .")[$"Determine whether" #h(0.2em) x = 5 #h(0.2em) "is a solution of" #h(0.2em) 6 x − 17 = 16 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image shows a linear algebraic equation: 6x - 17 = 16, typically solved for the variable 'x'. This equation represents a basic problem in algebra where one needs to isolate the variable.], alt: "The image shows a linear algebraic equation: 6x - 17 = 16, typically solved for the variable 'x'. This equation represents a basic problem in algebra where one needs to isolate the variable.", caption: none)]),
  [#figure(figph[The text 'Substitute 5 for x.' is displayed in a dark teal color, except for the number '5' which is highlighted in red. The background is white.], alt: "The text 'Substitute 5 for x.' is displayed in a dark teal color, except for the number '5' which is highlighted in red. The background is white.", caption: none)], [#figure(figph[A mathematical equation reads '6 x 5 - 17 ?= 16,' with the number '5' highlighted in red. The expression asks whether (6 times 5) minus 17 is equal to 16. 6x5=30, 30-17=13. 13 does not equal 16.], alt: "A mathematical equation reads '6 x 5 - 17 ?= 16,' with the number '5' highlighted in red. The expression asks whether (6 times 5) minus 17 is equal to 16. 6x5=30, 30-17=13. 13 does not equal 16.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical equation is displayed on a white background, reading '30 - 17 ?= 16'. The question mark is positioned directly above the equals sign, suggesting an inquiry into the equality of the two sides.], alt: "A mathematical equation is displayed on a white background, reading '30 - 17 ?= 16'. The question mark is positioned directly above the equals sign, suggesting an inquiry into the equality of the two sides.", caption: none)],
  [Subtract.], [#figure(figph[The image displays a mathematical inequality, '13 not equal to 16'.], alt: "The image displays a mathematical inequality, '13 not equal to 16'.", caption: none)],
))

So #math.equation(block: false, alt: "x equals 5")[$x = 5$] is not a solution to the equation #math.equation(block: false, alt: "6 x minus 17 equals 16 .")[$6 x − 17 = 16 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "Is x equals 3 a solution of 4 x minus 7 equals 16 ?")[$"Is" #h(0.2em) x = 3 #h(0.2em) "a solution of" #h(0.2em) 4 x − 7 = 16 "?"$]

#solutionbox[
no
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "Is x equals 2 a solution of 6 x minus 2 equals 10 ?")[$"Is" #h(0.2em) x = 2 #h(0.2em) "a solution of" #h(0.2em) 6 x − 2 = 10 "?"$]

#solutionbox[
yes
]
]

#examplebox("Example 2")[][
#math.equation(block: true, alt: "Determine whether y equals 2 is a solution of 6 y minus 4 equals 5 y minus 2 .")[$"Determine whether" #h(0.2em) y = 2 #h(0.2em) "is a solution of" #h(0.2em) 6 y − 4 = 5 y − 2 .$]

#solutionbox[
Here, the variable appears on both sides of the equation. We must substitute #math.equation(block: false, alt: "2")[$2$] for each #math.equation(block: false, alt: "y .")[$y .$] #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An algebraic equation is displayed, reading '6y - 4 = 5y - 2'.], alt: "An algebraic equation is displayed, reading '6y - 4 = 5y - 2'.", caption: none)]),
  [#figure(figph[The image displays the instruction 'Substitute 2 for y.' with the number 2 highlighted in red and the rest of the text in a blue-green color.], alt: "The image displays the instruction 'Substitute 2 for y.' with the number 2 highlighted in red and the rest of the text in a blue-green color.", caption: none)], [#figure(figph[A math problem asking whether 6(2) - 4 equals 5(2) - 2. Both sides simplify to 8, meaning the equation is true.], alt: "A math problem asking whether 6(2) - 4 equals 5(2) - 2. Both sides simplify to 8, meaning the equation is true.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical equation reads '12 - 4 ?= 10 - 2,' where the question mark indicates an unknown operator in a comparison between the two expressions. The solution would be 8 = 8, so the '?=' should be an equals sign.], alt: "A mathematical equation reads '12 - 4 ?= 10 - 2,' where the question mark indicates an unknown operator in a comparison between the two expressions. The solution would be 8 = 8, so the '?=' should be an equals sign.", caption: none)],
  [Subtract.], [#figure(figph[The image displays the equation '8 = 8' followed by a checkmark, indicating that the statement is mathematically correct.], alt: "The image displays the equation '8 = 8' followed by a checkmark, indicating that the statement is mathematically correct.", caption: none)],
))

Since #math.equation(block: false, alt: "y equals 2")[$y = 2$] results in a true equation, we know that #math.equation(block: false, alt: "2")[$2$] is a solution to the equation #math.equation(block: false, alt: "6 y minus 4 equals 5 y minus 2 .")[$6 y − 4 = 5 y − 2 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "Is y equals 3 a solution of 9 y minus 2 equals 8 y plus 1 ?")[$"Is" #h(0.2em) y = 3 #h(0.2em) "a solution of" #h(0.2em) 9 y − 2 = 8 y + 1 "?"$]

#solutionbox[
yes
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: true, alt: "Is y equals 4 a solution of 5 y minus 3 equals 3 y plus 5 ?")[$"Is" #h(0.2em) y = 4 #h(0.2em) "a solution of" #h(0.2em) 5 y − 3 = 3 y + 5 "?"$]

#solutionbox[
yes
]
]


=== Model the Subtraction Property of Equality

We will use a model to help you understand how the process of solving an equation is like solving a puzzle. An envelope represents the variable – since its contents are unknown – and each counter represents one.

Suppose a desk has an imaginary line dividing it in half. We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk as in . Both sides of the desk have the same number of counters, but some counters are hidden in the envelope. Can you tell how many counters are in the envelope?

#figure(figph[The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.", caption: none)

What steps are you taking in your mind to figure out how many counters are in the envelope? Perhaps you are thinking “I need to remove the #math.equation(block: false, alt: "3")[$3$] counters from the left side to get the envelope by itself. Those #math.equation(block: false, alt: "3")[$3$] counters on the left match with #math.equation(block: false, alt: "3")[$3$] on the right, so I can take them away from both sides. That leaves five counters on the right, so there must be #math.equation(block: false, alt: "5")[$5$] counters in the envelope.”  shows this process.

#figure(figph[The image is in two parts. On the left is a rectangle divided in half vertically. On the left side of the rectangle is an envelope with three counters below it. The 3 counters are circled in red with an arrow pointing out of the rectangle. On the right side is 8 counters. The bottom 3 counters are circled in red with an arrow pointing out of the rectangle. The 3 circled counters are removed from both sides of the rectangle, creating the new rectangle on the right of the image which is also divided in half vertically. On the left side of the rectangle is just an envelope. On the right side is 5 counters.], alt: "The image is in two parts. On the left is a rectangle divided in half vertically. On the left side of the rectangle is an envelope with three counters below it. The 3 counters are circled in red with an arrow pointing out of the rectangle. On the right side is 8 counters. The bottom 3 counters are circled in red with an arrow pointing out of the rectangle. The 3 circled counters are removed from both sides of the rectangle, creating the new rectangle on the right of the image which is also divided in half vertically. On the left side of the rectangle is just an envelope. On the right side is 5 counters.", caption: none)

What algebraic equation is modeled by this situation? Each side of the desk represents an #strong[expression] and the center line takes the place of the equal sign. We will call the contents of the envelope #math.equation(block: false, alt: "x ,")[$x ,$] so the number of counters on the left side of the desk is #math.equation(block: false, alt: "x plus 3 .")[$x + 3 .$] On the right side of the desk are #math.equation(block: false, alt: "8")[$8$] counters. We are told that #math.equation(block: false, alt: "x plus 3")[$x + 3$] is equal to #math.equation(block: false, alt: "8")[$8$] so our equation is#math.equation(block: false, alt: "x plus 3 equals 8 .")[$x + 3 = 8 .$]

#figure(figph[The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 8 counters.", caption: none)

#math.equation(block: true, alt: "x plus 3 equals 8")[$x + 3 = 8$]Let’s write algebraically the steps we took to discover how many counters were in the envelope.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing 'x + 3 = 8' in black text against a white background.], alt: "A mathematical equation is displayed, showing 'x + 3 = 8' in black text against a white background.", caption: none)]),
  [First, we took away three from each side.], [#figure(figph[Isolating 'x' in x + 3 = 8 by subtracting 3 from both sides of the equation, shown as x + 3 - 3 = 8 - 3, with the subtracted 3s highlighted in red.], alt: "Isolating 'x' in x + 3 = 8 by subtracting 3 from both sides of the equation, shown as x + 3 - 3 = 8 - 3, with the subtracted 3s highlighted in red.", caption: none)],
  [Then we were left with five.], [#figure(figph[The equation x = 5 is displayed in black text on a white background.], alt: "The equation x = 5 is displayed in black text on a white background.", caption: none)],
))

Now let’s check our solution. We substitute #math.equation(block: false, alt: "5")[$5$] for #math.equation(block: false, alt: "x")[$x$] in the original equation and see if we get a true statement.

#figure(figph[The image shows the original equation, x plus 3 equal to 8. Substitute 5 in for x to check. The equation becomes 5 plus 3 equal to 8. Is this true? The left side simplifies by adding 5 and 3 to get 8. Both sides of the equal symbol are 8.], alt: "The image shows the original equation, x plus 3 equal to 8. Substitute 5 in for x to check. The equation becomes 5 plus 3 equal to 8. Is this true? The left side simplifies by adding 5 and 3 to get 8. Both sides of the equal symbol are 8.", caption: none)

Our solution is correct. Five counters in the envelope plus three more equals eight.

#examplebox("Example 3")[][
Write an equation modeled by the envelopes and counters, and then solve the equation:

#figure(figph[The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 5 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 5 counters.", caption: none)

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([On the left, write #math.equation(block: false, alt: "x")[$x$] for the contents of the envelope, add the #math.equation(block: false, alt: "4")[$4$] counters, so we have #math.equation(block: false, alt: "x plus 4")[$x + 4$].], [#math.equation(block: false, alt: "x plus 4")[$x + 4$]]),
  [On the right, there are #math.equation(block: false, alt: "5")[$5$] counters.], [#math.equation(block: false, alt: "5")[$5$]],
  [The two sides are equal.], [#math.equation(block: false, alt: "x plus 4 equals 5")[$x + 4 = 5$]],
  [Solve the equation by subtracting #math.equation(block: false, alt: "4")[$4$] counters from each side.], [],
))

#figure(figph[The image is in two parts. On the left is a rectangle divided in half vertically. On the left side of the rectangle is an envelope with 4 counters below it. The 4 counters are circled in red with an arrow pointing out of the rectangle. On the right side is 5 counters. The bottom 4 counters are circled in red with an arrow pointing out of the rectangle. The 4 circled counters are removed from both sides of the rectangle, creating the new rectangle on the right of the image which is also divided in half vertically. On the left side of the rectangle is just an envelope. On the right side is 1 counter.], alt: "The image is in two parts. On the left is a rectangle divided in half vertically. On the left side of the rectangle is an envelope with 4 counters below it. The 4 counters are circled in red with an arrow pointing out of the rectangle. On the right side is 5 counters. The bottom 4 counters are circled in red with an arrow pointing out of the rectangle. The 4 circled counters are removed from both sides of the rectangle, creating the new rectangle on the right of the image which is also divided in half vertically. On the left side of the rectangle is just an envelope. On the right side is 1 counter.", caption: none)

We can see that there is one counter in the envelope. This can be shown algebraically as: #linebreak() 
#figure(figph[The image shows the given equation, x plus 4 equal to 5. Take 4 away from both sides of the equation to get x plus 4 minus 4 equal to 5 minus 4. On the left, plus 4 and minus 4 cancel out to leave just x. On the right 5 minus 4 is 1. The equation becomes x equal to 1.], alt: "The image shows the given equation, x plus 4 equal to 5. Take 4 away from both sides of the equation to get x plus 4 minus 4 equal to 5 minus 4. On the left, plus 4 and minus 4 cancel out to leave just x. On the right 5 minus 4 is 1. The equation becomes x equal to 1.", caption: none)

Substitute #math.equation(block: false, alt: "1")[$1$] for #math.equation(block: false, alt: "x")[$x$] in the equation to check. #linebreak() 
#figure(figph[The image shows the original equation, x plus 4 equal to 5. Substitute 1 in for x to check. The equation becomes 1 plus 4 equal to 5. Is this true? The left side simplifies by adding 1 and 4 to get 5. Both sides of the equal symbol are 5.], alt: "The image shows the original equation, x plus 4 equal to 5. Substitute 1 in for x to check. The equation becomes 1 plus 4 equal to 5. Is this true? The left side simplifies by adding 1 and 4 to get 5. Both sides of the equal symbol are 5.", caption: none)

Since #math.equation(block: false, alt: "x equals 1")[$x = 1$] makes the statement true, we know that #math.equation(block: false, alt: "1")[$1$] is indeed a solution.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the equation modeled by the envelopes and counters, and then solve the equation:

#figure(figph[The image is divided in half vertically. On the left side is an envelope with one counter below it. On the right side is 7 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with one counter below it. On the right side is 7 counters.", caption: none)

#solutionbox[
#emph[x] + 1 = 7; #emph[x] = 6
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write the equation modeled by the envelopes and counters, and then solve the equation:

#figure(figph[The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 4 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 4 counters.", caption: none)

#solutionbox[
#emph[x] + 3 = 4; #emph[x] = 1
]
]


=== Solve Equations Using the Subtraction Property of Equality

Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equations. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Subtraction Property of Equality]

For any numbers #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c ,")[$c ,$] if

#math.equation(block: true, alt: "a equals b")[$a = b$]then

#math.equation(block: true, alt: "a minus c equals b minus c")[$a − c = b − c$]
]

Think about twin brothers Andy and Bobby. They are #math.equation(block: false, alt: "17")[$17$] years old. How old was Andy #math.equation(block: false, alt: "3")[$3$] years ago? He was #math.equation(block: false, alt: "3")[$3$] years less than #math.equation(block: false, alt: "17 ,")[$17 ,$] so his age was #math.equation(block: false, alt: "17 minus 3 ,")[$17 − 3 ,$] or #math.equation(block: false, alt: "14 .")[$14 .$] What about Bobby’s age #math.equation(block: false, alt: "3")[$3$] years ago? Of course, he was #math.equation(block: false, alt: "14")[$14$] also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages #math.equation(block: false, alt: "3")[$3$] years ago.

#math.equation(block: true, alt: "a equals b; a minus 3 equals b minus 3")[$a = b \ a − 3 = b − 3$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve an equation using the Subtraction Property of Equality.]

+ Use the Subtraction Property of Equality to isolate the variable.
+ Simplify the expressions on both sides of the equation.
+ Check the solution.
]

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "x plus 8 equals 17 .")[$x + 8 = 17 .$]

#solutionbox[
We will use the Subtraction Property of Equality to isolate #math.equation(block: false, alt: "x .")[$x .$] #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An algebraic equation is shown, displaying 'x + 8 = 17' in black text on a white background.], alt: "An algebraic equation is shown, displaying 'x + 8 = 17' in black text on a white background.", caption: none)]),
  [Subtract 8 from both sides.], [#figure(figph[The equation x + 8 - 8 = 17 - 8, demonstrating the subtraction property of equality to isolate the variable x.], alt: "The equation x + 8 - 8 = 17 - 8, demonstrating the subtraction property of equality to isolate the variable x.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation, x = 9, displayed in black text on a white background.], alt: "A mathematical equation, x = 9, displayed in black text on a white background.", caption: none)],
  [], [#figure(figph[A basic algebraic equation is displayed, showing 'x + 8 = 17' in black text against a white background. This simple addition equation requires solving for the variable 'x'.], alt: "A basic algebraic equation is displayed, showing 'x + 8 = 17' in black text against a white background. This simple addition equation requires solving for the variable 'x'.", caption: none)],
  [], [#figure(figph[A mathematical equation shows '9 + 8 = 17'. The number '9' is red, while the rest of the equation is black.], alt: "A mathematical equation shows '9 + 8 = 17'. The number '9' is red, while the rest of the equation is black.", caption: none)],
  [], [#figure(figph[The image shows the equation '17 = 17' followed by a checkmark, indicating that the statement is correct and verified.], alt: "The image shows the equation '17 = 17' followed by a checkmark, indicating that the statement is correct and verified.", caption: none)],
))

Since #math.equation(block: false, alt: "x equals 9")[$x = 9$] makes #math.equation(block: false, alt: "x plus 8 equals 17")[$x + 8 = 17$] a true statement, we know #math.equation(block: false, alt: "9")[$9$] is the solution to the equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "x plus 6 equals 19")[$x + 6 = 19$]

#solutionbox[
#emph[x] = 13
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "x plus 9 equals 14")[$x + 9 = 14$]

#solutionbox[
#emph[x] = 5
]
]

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "100 equals y plus 74 .")[$100 = y + 74 .$]

#solutionbox[
To solve an equation, we must always isolate the variable—it doesn’t matter which side it is on. To isolate #math.equation(block: false, alt: "y ,")[$y ,$] we will subtract #math.equation(block: false, alt: "74")[$74$] from both sides. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing '100 = y + 74' in black text against a white background.], alt: "A mathematical equation is displayed, showing '100 = y + 74' in black text against a white background.", caption: none)]),
  [Subtract 74 from both sides.], [#figure(figph[A mathematical equation displays '100 - 74 = y + 74 - 74', illustrating the subtraction property of equality where 74 is subtracted from both sides to isolate the variable 'y'.], alt: "A mathematical equation displays '100 - 74 = y + 74 - 74', illustrating the subtraction property of equality where 74 is subtracted from both sides to isolate the variable 'y'.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation '26 = y' in a clean, legible font against a plain white background, presenting a simple assignment of the value 26 to the variable y.], alt: "The image displays a mathematical equation '26 = y' in a clean, legible font against a plain white background, presenting a simple assignment of the value 26 to the variable y.", caption: none)],
  [Substitute #math.equation(block: false, alt: "26")[$26$] for #math.equation(block: false, alt: "y")[$y$] to check. #linebreak() 
#figure(figph[The image shows the original equation,100 equal to y plus 74. Substitute 26 in for y to check. The equation becomes 100 equal to 26 plus 74. Is this true? The right side simplifies by adding 26 and 74 to get 100. Both sides of the equal symbol are 100.], alt: "The image shows the original equation,100 equal to y plus 74. Substitute 26 in for y to check. The equation becomes 100 equal to 26 plus 74. Is this true? The right side simplifies by adding 26 and 74 to get 100. Both sides of the equal symbol are 100.", caption: none)], [],
))

Since #math.equation(block: false, alt: "y equals 26")[$y = 26$] makes #math.equation(block: false, alt: "100 equals y plus 74")[$100 = y + 74$] a true statement, we have found the solution to this equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "95 equals y plus 67")[$95 = y + 67$]

#solutionbox[
#emph[y] = 28
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "91 equals y plus 45")[$91 = y + 45$]

#solutionbox[
#emph[y] = 46
]
]


=== Solve Equations Using the Addition Property of Equality

In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.

But suppose we have an equation with a number subtracted from the variable, such as #math.equation(block: false, alt: "x minus 5 equals 8 .")[$x − 5 = 8 .$] We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides.

We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Addition Property of Equality]

For any numbers #math.equation(block: false, alt: "a , b")[$a , b$], and #math.equation(block: false, alt: "c")[$c$], if

#math.equation(block: true, alt: "a equals b")[$a = b$]then

#math.equation(block: true, alt: "a plus c equals b plus c")[$a + c = b + c$]
]

Remember the #math.equation(block: false, alt: "17 -year-old")[$17 "-year-old"$] twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be #math.equation(block: false, alt: "27 .")[$27 .$]

#math.equation(block: true, alt: "a equals b; a plus 10 equals b plus 10")[$a = b \ a + 10 = b + 10$]We can add the same number to both sides and still keep the equality.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve an equation using the Addition Property of Equality.]

+ Use the Addition Property of Equality to isolate the variable.
+ Simplify the expressions on both sides of the equation.
+ Check the solution.
]

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "x minus 5 equals 8 .")[$x − 5 = 8 .$]

#solutionbox[
We will use the #strong[Addition Property of Equality] to isolate the variable. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A simple algebraic equation is displayed, showing 'x - 5 = 8' in a dark, bold font on a white background.], alt: "A simple algebraic equation is displayed, showing 'x - 5 = 8' in a dark, bold font on a white background.", caption: none)]),
  [Add 5 to both sides.], [#figure(figph[A mathematical equation shows 'x minus 5 plus 5 equals 8 plus 5', demonstrating the addition property of equality where 5 is added to both sides of the equation x - 5 = 8.], alt: "A mathematical equation shows 'x minus 5 plus 5 equals 8 plus 5', demonstrating the addition property of equality where 5 is added to both sides of the equation x - 5 = 8.", caption: none)],
  [Simplify.], [#figure(figph[A simple mathematical equation is displayed on a white background, showing 'x = 13'.], alt: "A simple mathematical equation is displayed on a white background, showing 'x = 13'.", caption: none)],
  [#figure(figph[The image displays the text 'Now we can check. Let x = 13.', suggesting a step in a mathematical or problem-solving process where a value is substituted for a variable to verify a solution.], alt: "The image displays the text 'Now we can check. Let x = 13.', suggesting a step in a mathematical or problem-solving process where a value is substituted for a variable to verify a solution.", caption: none)], [],
  [#figure(figph[A mathematical equation, rendered in black text on a white background, shows 'x - 5 = 8'.], alt: "A mathematical equation, rendered in black text on a white background, shows 'x - 5 = 8'.", caption: none)], [],
  [#figure(figph[A mathematical equation shows '13 - 5 =? 8', with the question mark above the equals sign. This expression tests if thirteen minus five is indeed equal to eight, which is a true statement as 13 - 5 equals 8.], alt: "A mathematical equation shows '13 - 5 =? 8', with the question mark above the equals sign. This expression tests if thirteen minus five is indeed equal to eight, which is a true statement as 13 - 5 equals 8.", caption: none)], [],
  [#figure(figph[The mathematical statement '8 = 8' is shown, accompanied by a checkmark, indicating its correctness or verification.], alt: "The mathematical statement '8 = 8' is shown, accompanied by a checkmark, indicating its correctness or verification.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "x minus 9 equals 13")[$x − 9 = 13$]

#solutionbox[
#emph[x] = 22
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

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

#solutionbox[
#emph[y] = 4
]
]

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "27 equals a minus 16 .")[$27 = a − 16 .$]

#solutionbox[
We will add #math.equation(block: false, alt: "16")[$16$] to each side to isolate the variable. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The image displays the mathematical equation 27 = a - 16, centered against a white background. This equation represents a basic algebraic problem where the variable 'a' needs to be solved.], alt: "The image displays the mathematical equation 27 = a - 16, centered against a white background. This equation represents a basic algebraic problem where the variable 'a' needs to be solved.", caption: none)]),
  [Add 16 to each side.], [#figure(figph[An algebraic equation showing 27 + 16 = a - 16 + 16, with plus signs and the number 16 on the right side highlighted in red against a white background.], alt: "An algebraic equation showing 27 + 16 = a - 16 + 16, with plus signs and the number 16 on the right side highlighted in red against a white background.", caption: none)],
  [Simplify.], [#figure(figph[The equation 43 = a is displayed on a white background, indicating that the value of the variable 'a' is 43.], alt: "The equation 43 = a is displayed on a white background, indicating that the value of the variable 'a' is 43.", caption: none)],
  [#figure(figph[The text 'Now we can check. Let q = 43.' is displayed on a white background. The number 43 is highlighted in red, while the rest of the text is in a dark blue-grey color.], alt: "The text 'Now we can check. Let q = 43.' is displayed on a white background. The number 43 is highlighted in red, while the rest of the text is in a dark blue-grey color.", caption: none)], [#figure(figph[A basic algebraic equation is displayed, showing '27 = a - 16' against a plain white background. The equation requires solving for the variable 'a'.], alt: "A basic algebraic equation is displayed, showing '27 = a - 16' against a plain white background. The equation requires solving for the variable 'a'.", caption: none)],
  [], [#figure(figph[A math problem displaying '27 ?= 43 - 16' with a question mark above the equals sign, indicating a query about the equality of the two sides of the equation. The number 43 is highlighted in red.], alt: "A math problem displaying '27 ?= 43 - 16' with a question mark above the equals sign, indicating a query about the equality of the two sides of the equation. The number 43 is highlighted in red.", caption: none)],
  [], [#figure(figph[The number 27 is shown to be equal to 27, followed by a checkmark, indicating correctness or agreement.], alt: "The number 27 is shown to be equal to 27, followed by a checkmark, indicating correctness or agreement.", caption: none)],
))

The solution to #math.equation(block: false, alt: "27 equals a minus 16")[$27 = a − 16$] is #math.equation(block: false, alt: "a equals 43 .")[$a = 43 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "19 equals a minus 18")[$19 = a − 18$]

#solutionbox[
#emph[a] = 37
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve:

#math.equation(block: true, alt: "27 equals n minus 14")[$27 = n − 14$]

#solutionbox[
#emph[n] = 41
]
]


=== Translate Word Phrases to Algebraic Equations

Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean #emph[equals]. Some words that translate to the equal sign are:

- is equal to
- is the same as
- is
- gives
- was
- will be

It may be helpful to put a box around the #emph[equals] word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.

We will practice translating word sentences into algebraic equations. Some of the sentences will be basic number facts with no variables to solve for. Some sentences will translate into equations with variables. The focus right now is just to translate the words into algebra.

#examplebox("Example 8")[][
Translate the sentence into an algebraic equation: The sum of #math.equation(block: false, alt: "6")[$6$] and #math.equation(block: false, alt: "9")[$9$] is #math.equation(block: false, alt: "15 .")[$15 .$]

#solutionbox[
The word #emph[is] tells us the equal sign goes between 9 and 15. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Locate the “equals” word(s).], [#figure(figph[Illustrates replacing 'is' with '='. The statement 'The sum of 6 and 9 is 15' becomes the mathematical expression 'The sum of 6 and 9 = 15'.], alt: "Illustrates replacing 'is' with '='. The statement 'The sum of 6 and 9 is 15' becomes the mathematical expression 'The sum of 6 and 9 = 15'.", caption: none)]),
  [Write the = sign.], [],
  [Translate the words to the left of the #emph[equals] word into an algebraic expression.], [#figure(figph[A simple math problem is displayed on a white background, showing the equation '6 + 9 = \_\_\_' with a blank space for the answer, indicating an addition sum to be solved.], alt: "A simple math problem is displayed on a white background, showing the equation '6 + 9 = ___' with a blank space for the answer, indicating an addition sum to be solved.", caption: none)],
  [Translate the words to the right of the #emph[equals] word into an algebraic expression.], [#figure(figph[The equation 6 + 9 = 15 is displayed in a dark blue font against a white background.], alt: "The equation 6 + 9 = 15 is displayed in a dark blue font against a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the sentence into an algebraic equation:

The sum of #math.equation(block: false, alt: "7")[$7$] and #math.equation(block: false, alt: "6")[$6$] gives #math.equation(block: false, alt: "13 .")[$13 .$]

#solutionbox[
7 + 6 = 13
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the sentence into an algebraic equation:

The sum of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "6")[$6$] is #math.equation(block: false, alt: "14 .")[$14 .$]

#solutionbox[
8 + 6 = 14
]
]

#examplebox("Example 9")[][
Translate the sentence into an algebraic equation: The product of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "7")[$7$] is #math.equation(block: false, alt: "56 .")[$56 .$]

#solutionbox[
The location of the word #emph[is] tells us that the equal sign goes between 7 and 56. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Locate the “equals” word(s).], [#figure(figph[This image demonstrates how the word 'is' in a descriptive mathematical statement, like 'The product of 8 and 7 is 56,' is equivalent to the equals sign (=), shown as 'The product of 8 and 7 = 56.'], alt: "This image demonstrates how the word 'is' in a descriptive mathematical statement, like 'The product of 8 and 7 is 56,' is equivalent to the equals sign (=), shown as 'The product of 8 and 7 = 56.'", caption: none)]),
  [Write the = sign.], [],
  [Translate the words to the left of the #emph[equals] word into an algebraic expression.], [#figure(figph[A mathematical equation is shown with '8 . 7 = \_' on a white background, indicating a multiplication problem where the product of 8 and 7 needs to be filled in.], alt: "A mathematical equation is shown with '8 . 7 = _' on a white background, indicating a multiplication problem where the product of 8 and 7 needs to be filled in.", caption: none)],
  [Translate the words to the right of the #emph[equals] word into an algebraic expression.], [#figure(figph[The image displays the multiplication equation 8 multiplied by 7 equals 56, written as '8 ', '.', ' 7 = 56' in a dark blue or gray font against a white background.], alt: "The image displays the multiplication equation 8 multiplied by 7 equals 56, written as '8 ', '.', ' 7 = 56' in a dark blue or gray font against a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the sentence into an algebraic equation:

The product of #math.equation(block: false, alt: "6")[$6$] and #math.equation(block: false, alt: "9")[$9$] is #math.equation(block: false, alt: "54 .")[$54 .$]

#solutionbox[
6 ⋅ 9 = 54
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the sentence into an algebraic equation:

The product of #math.equation(block: false, alt: "21")[$21$] and #math.equation(block: false, alt: "3")[$3$] gives #math.equation(block: false, alt: "63 .")[$63 .$]

#solutionbox[
21 ⋅ 3 = 63
]
]

#examplebox("Example 10")[][
Translate the sentence into an algebraic equation: Twice the difference of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "3")[$3$] gives #math.equation(block: false, alt: "18 .")[$18 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Locate the “equals” word(s).], [#figure(figph[The text 'Twice the difference of x and 3 gives 18.' is displayed on a white background, with the word 'gives' enclosed in a thin black rectangle.], alt: "The text 'Twice the difference of x and 3 gives 18.' is displayed on a white background, with the word 'gives' enclosed in a thin black rectangle.", caption: none)]),
  [Recognize the key words: #emph[twice; difference of …. and …].], [#emph[Twice] means two times.],
  [Translate.], [#figure(figph[This image translates the word problem 'Twice the difference of x and 3 gives 18' into its algebraic equation form, '2(x - 3) = 18', highlighting corresponding parts.], alt: "This image translates the word problem 'Twice the difference of x and 3 gives 18' into its algebraic equation form, '2(x - 3) = 18', highlighting corresponding parts.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the given sentence into an algebraic equation:

Twice the difference of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "5")[$5$] gives #math.equation(block: false, alt: "30 .")[$30 .$]

#solutionbox[
2(#emph[x] − 5) = 30
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate the given sentence into an algebraic equation:

Twice the difference of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "4")[$4$] gives #math.equation(block: false, alt: "16 .")[$16 .$]

#solutionbox[
2(#emph[y] − 4) = 16
]
]


=== Translate to an Equation and Solve

Now let’s practice translating sentences into algebraic equations and then solving them. We will solve the equations by using the Subtraction and Addition Properties of Equality.

#examplebox("Example 11")[][
Translate and solve: Three more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "47 .")[$47 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [Three more than #emph[x] is equal to 47.], [],
  [Translate.], [#figure(figph[A simple algebraic equation is displayed, showing 'x + 3 = 47' in a bold, sans-serif font against a white background.], alt: "A simple algebraic equation is displayed, showing 'x + 3 = 47' in a bold, sans-serif font against a white background.", caption: none)], [],
  [Subtract 3 from both sides of the equation.], [#figure(figph[A mathematical equation shows 'x + 3 - 3 = 47 - 3' with the second '3' on the left side and the '3' on the right side of the equation highlighted in red to emphasize subtraction from both sides.], alt: "A mathematical equation shows 'x + 3 - 3 = 47 - 3' with the second '3' on the left side and the '3' on the right side of the equation highlighted in red to emphasize subtraction from both sides.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation 'x = 44' is displayed in black text against a white background.], alt: "The mathematical equation 'x = 44' is displayed in black text against a white background.", caption: none)], [],
  [We can check. Let #math.equation(block: false, alt: "x equals 44")[$x = 44$].], [#figure(figph[A mathematical equation is displayed with a variable 'x', showing 'x + 3 = 47' against a plain white background.], alt: "A mathematical equation is displayed with a variable 'x', showing 'x + 3 = 47' against a plain white background.", caption: none)], [],
  [], [#figure(figph[An equation displaying 44 + 3 with a question mark over the equals sign before 47, asking for verification.], alt: "An equation displaying 44 + 3 with a question mark over the equals sign before 47, asking for verification.", caption: none)], [],
  [], [#figure(figph[The number 47 is shown equal to 47, followed by a checkmark, indicating the mathematical statement's correctness or verification.], alt: "The number 47 is shown equal to 47, followed by a checkmark, indicating the mathematical statement's correctness or verification.", caption: none)], [],
))

So #math.equation(block: false, alt: "x equals 44")[$x = #h(0.2em) 44$] is the solution.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

Seven more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "37 .")[$37 .$]

#solutionbox[
#emph[x] + 7 = 37; #emph[x] = 30
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

Eleven more than #math.equation(block: false, alt: "y")[$y$] is equal to #math.equation(block: false, alt: "28 .")[$28 .$]

#solutionbox[
#emph[y] + 11 = 28; #emph[y] = 17
]
]

#examplebox("Example 12")[][
Translate and solve: The difference of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "14")[$14$] is #math.equation(block: false, alt: "18 .")[$18 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [The difference of #emph[y] and 14 is 18.], [],
  [Translate.], [#figure(figph[A mathematical equation is displayed, showing 'y - 14 = 18' in black text on a white background. This equation represents a basic algebraic problem where 'y' is an unknown variable.], alt: "A mathematical equation is displayed, showing 'y - 14 = 18' in black text on a white background. This equation represents a basic algebraic problem where 'y' is an unknown variable.", caption: none)], [],
  [Add 14 to both sides.], [#figure(figph[A mathematical equation showing y - 14 + 14 = 18 + 14, with the number 14 highlighted in red as it is added to both sides of the equation.], alt: "A mathematical equation showing y - 14 + 14 = 18 + 14, with the number 14 highlighted in red as it is added to both sides of the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows the mathematical equation y = 32 in black text on a white background. The variable 'y' is set equal to the numerical value '32', indicating a simple assignment or a constant.], alt: "The image shows the mathematical equation y = 32 in black text on a white background. The variable 'y' is set equal to the numerical value '32', indicating a simple assignment or a constant.", caption: none)], [],
  [We can check. Let #math.equation(block: false, alt: "y equals 32")[$y = 32$].], [#figure(figph[A mathematical equation is displayed, showing 'y - 14 = 18' in black text on a white background. This is a basic algebraic problem to solve for the variable 'y'.], alt: "A mathematical equation is displayed, showing 'y - 14 = 18' in black text on a white background. This is a basic algebraic problem to solve for the variable 'y'.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows 32 minus 14 followed by an equals sign with a question mark above it, and then the number 18, asking if 32 - 14 equals 18.], alt: "A mathematical equation shows 32 minus 14 followed by an equals sign with a question mark above it, and then the number 18, asking if 32 - 14 equals 18.", caption: none)], [],
  [], [#figure(figph[The image shows a mathematical expression '18 = 18' followed by a checkmark, symbolizing correctness or validation of the equality.], alt: "The image shows a mathematical expression '18 = 18' followed by a checkmark, symbolizing correctness or validation of the equality.", caption: none)], [],
))

So #math.equation(block: false, alt: "y equals 32")[$y = 32$] is the solution.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

The difference of #math.equation(block: false, alt: "z")[$z$] and #math.equation(block: false, alt: "17")[$17$] is equal to #math.equation(block: false, alt: "37 .")[$37 .$]

#solutionbox[
#emph[z] − 17 = 37; #emph[z] = 54
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

The difference of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "19")[$19$] is equal to #math.equation(block: false, alt: "45 .")[$45 .$]

#solutionbox[
#emph[x] − 19 = 45; #emph[x] = 64
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://openstaxcollege.org/l/24Solveonestep")[Solving One Step Equations By Addition and Subtraction]
]


=== Key Concepts

- #emph[Determine whether a number is a solution to an equation.]
+ Substitute the number for the variable in the equation.
+ Simplify the expressions on both sides of the equation.
+ Determine whether the resulting equation is true.  If it is true, the number is a solution.


           If it is not true, the number is not a solution.
- #emph[Subtraction Property of Equality]
- For any numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and  #math.equation(block: false, alt: "c")[$c$],#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([if], [#math.equation(block: false, alt: "a equals b")[$a = b$]]),
  [then], [#math.equation(block: false, alt: "a minus c equals b minus c")[$a − c = b − c$]],
))
- #emph[Solve an equation using the Subtraction Property of Equality.]
+ Use the Subtraction Property of Equality to isolate the variable.
+ Simplify the expressions on both sides of the equation.
+ Check the solution.
- #emph[Addition Property of Equality]
- For any numbers #math.equation(block: false, alt: "a")[$a$], #math.equation(block: false, alt: "b")[$b$], and  #math.equation(block: false, alt: "c")[$c$],#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([if], [#math.equation(block: false, alt: "a equals b")[$a = b$]]),
  [then], [#math.equation(block: false, alt: "a plus c equals b plus c")[$a + c = b + c$]],
))
- #emph[Solve an equation using the Addition Property of Equality.]
+ Use the Addition Property of Equality to isolate the variable.
+ Simplify the expressions on both sides of the equation.
+ Check the solution.


==== Practice Makes Perfect

#strong[Determine Whether a Number is a Solution of an Equation]

In the following exercises, determine whether each given value is a solution to the equation.

#math.equation(block: true, alt: "x plus 13 equals 21")[$x + 13 = 21$]

+ ⓐ #math.equation(block: false, alt: "x equals 8")[$#h(0.2em) x = 8$]
+ ⓑ #math.equation(block: false, alt: "x equals 34")[$#h(0.2em) x = 34$]

#solutionbox[
+ ⓐ yes
+ ⓑ no
]

#math.equation(block: true, alt: "y plus 18 equals 25")[$y + 18 = 25$]

+ ⓐ #math.equation(block: false, alt: "y equals 7")[$#h(0.2em) y = 7$]
+ ⓑ #math.equation(block: false, alt: "y equals 43")[$y = 43$]

#math.equation(block: true, alt: "m minus 4 equals 13")[$m − 4 = 13$]

+ ⓐ #math.equation(block: false, alt: "m equals 9")[$#h(0.2em) m = 9$]
+ ⓑ #math.equation(block: false, alt: "m equals 17")[$m = 17$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
]

#math.equation(block: true, alt: "n minus 9 equals 6")[$n − 9 = 6$]

+ ⓐ #math.equation(block: false, alt: "n equals 3")[$#h(0.2em) n = 3$]
+ ⓑ #math.equation(block: false, alt: "n equals 15")[$n = 15$]

#math.equation(block: true, alt: "3 p plus 6 equals 15")[$3 p + 6 = 15$]

+ ⓐ #math.equation(block: false, alt: "p equals 3")[$#h(0.2em) p = 3$]
+ ⓑ #math.equation(block: false, alt: "p equals 7")[$#h(0.2em) p = 7$]

#solutionbox[
+ ⓐ yes
+ ⓑ no
]

#math.equation(block: true, alt: "8 q plus 4 equals 20")[$8 q + 4 = 20$]

+ ⓐ #math.equation(block: false, alt: "q equals 2")[$#h(0.2em) q = 2$]
+ ⓑ #math.equation(block: false, alt: "q equals 3")[$#h(0.2em) q = 3$]

#math.equation(block: true, alt: "18 d minus 9 equals 27")[$18 d − 9 = 27$]

+ ⓐ #math.equation(block: false, alt: "d equals 1")[$#h(0.2em) d = 1$]
+ ⓑ #math.equation(block: false, alt: "d equals 2")[$#h(0.2em) d = 2$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
]

#math.equation(block: true, alt: "24 f minus 12 equals 60")[$24 f − 12 = 60$]

+ ⓐ #math.equation(block: false, alt: "f equals 2")[$#h(0.2em) f = 2$]
+ ⓑ #math.equation(block: false, alt: "f equals 3")[$#h(0.2em) f = 3$]

#math.equation(block: true, alt: "8 u minus 4 equals 4 u plus 40")[$8 u − 4 = 4 u + 40$]

+ ⓐ #math.equation(block: false, alt: "u equals 3")[$#h(0.2em) u = 3$]
+ ⓑ #math.equation(block: false, alt: "u equals 11")[$u = 11$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
]

#math.equation(block: true, alt: "7 v minus 3 equals 4 v plus 36")[$7 v − 3 = 4 v + 36$]

+ ⓐ #math.equation(block: false, alt: "v equals 3")[$#h(0.2em) v = 3$]
+ ⓑ #math.equation(block: false, alt: "v equals 11")[$v = 11$]

#math.equation(block: true, alt: "20 h minus 5 equals 15 h plus 35")[$20 h − 5 = 15 h + 35$]

+ ⓐ #math.equation(block: false, alt: "h equals 6")[$#h(0.2em) h = 6$]
+ ⓑ #math.equation(block: false, alt: "h equals 8")[$h = 8$]

#solutionbox[
+ ⓐ no
+ ⓑ yes
]

#math.equation(block: true, alt: "18 k minus 3 equals 12 k plus 33")[$18 k − 3 = 12 k + 33$]

+ ⓐ #math.equation(block: false, alt: "k equals 1")[$k = 1$]
+ ⓑ #math.equation(block: false, alt: "k equals 6")[$k = 6$]

#strong[Model the Subtraction Property of Equality]

In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality.

#figure(figph[The image is divided in half vertically. On the left side is an envelope with 2 counters below it. On the right side is 5 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with 2 counters below it. On the right side is 5 counters.", caption: none)

#solutionbox[
#emph[x] + 2 = 5; #emph[x] = 3
]

#figure(figph[The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 7 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 7 counters.", caption: none)

#figure(figph[The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 6 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 6 counters.", caption: none)

#solutionbox[
#emph[x] + 3 = 6; #emph[x] = 3
]

#figure(figph[The image is divided in half vertically. On the left side is an envelope with 5 counters below it. On the right side is 9 counters.], alt: "The image is divided in half vertically. On the left side is an envelope with 5 counters below it. On the right side is 9 counters.", caption: none)

#strong[Solve Equations using the Subtraction Property of Equality]

In the following exercises, solve each equation using the subtraction property of equality.

#math.equation(block: true, alt: "a plus 2 equals 18")[$a + 2 = 18$]

#solutionbox[
#emph[a] = 16
]

#math.equation(block: true, alt: "b plus 5 equals 13")[$b + 5 = 13$]

#math.equation(block: true, alt: "p plus 18 equals 23")[$p + 18 = 23$]

#solutionbox[
#emph[p] = 5
]

#math.equation(block: true, alt: "q plus 14 equals 31")[$q + 14 = 31$]

#math.equation(block: true, alt: "r plus 76 equals 100")[$r + 76 = 100$]

#solutionbox[
#emph[r] = 24
]

#math.equation(block: true, alt: "s plus 62 equals 95")[$s + 62 = 95$]

#math.equation(block: true, alt: "16 equals x plus 9")[$16 = x + 9$]

#solutionbox[
#emph[x] = 7
]

#math.equation(block: true, alt: "17 equals y plus 6")[$17 = y + 6$]

#math.equation(block: true, alt: "93 equals p plus 24")[$93 = p + 24$]

#solutionbox[
#emph[p] = 69
]

#math.equation(block: true, alt: "116 equals q plus 79")[$116 = q + 79$]

#math.equation(block: true, alt: "465 equals d plus 398")[$465 = d + 398$]

#solutionbox[
#emph[d] = 67
]

#math.equation(block: true, alt: "932 equals c plus 641")[$932 = c + 641$]

#strong[Solve Equations using the Addition Property of Equality]

In the following exercises, solve each equation using the addition property of equality.

#math.equation(block: true, alt: "y minus 3 equals 19")[$y − 3 = 19$]

#solutionbox[
#emph[y] = 22
]

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

#math.equation(block: true, alt: "u minus 6 equals 24")[$u − 6 = 24$]

#solutionbox[
#emph[u] = 30
]

#math.equation(block: true, alt: "v minus 7 equals 35")[$v − 7 = 35$]

#math.equation(block: true, alt: "f minus 55 equals 123")[$f − 55 = 123$]

#solutionbox[
#emph[f] = 178
]

#math.equation(block: true, alt: "g minus 39 equals 117")[$g − 39 = 117$]

#math.equation(block: true, alt: "19 equals n minus 13")[$19 = n − 13$]

#solutionbox[
#emph[n] = 32
]

#math.equation(block: true, alt: "18 equals m minus 15")[$18 = m − 15$]

#math.equation(block: true, alt: "10 equals p minus 38")[$10 = p − 38$]

#solutionbox[
#emph[p] = 48
]

#math.equation(block: true, alt: "18 equals q minus 72")[$18 = q − 72$]

#math.equation(block: true, alt: "268 equals y minus 199")[$268 = y − 199$]

#solutionbox[
#emph[y] = 467
]

#math.equation(block: true, alt: "204 equals z minus 149")[$204 = z − 149$]

#strong[Translate Word Phrase to Algebraic Equations]

In the following exercises, translate the given sentence into an algebraic equation.

The sum of #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "9")[$9$] is equal to #math.equation(block: false, alt: "17 .")[$17 .$]

#solutionbox[
8 + 9 = 17
]

The sum of #math.equation(block: false, alt: "7")[$7$] and #math.equation(block: false, alt: "9")[$9$] is equal to #math.equation(block: false, alt: "16 .")[$16 .$]

The difference of #math.equation(block: false, alt: "23")[$23$] and #math.equation(block: false, alt: "19")[$19$] is equal to #math.equation(block: false, alt: "4 .")[$4 .$]

#solutionbox[
23 − 19 = 4
]

The difference of #math.equation(block: false, alt: "29")[$29$] and #math.equation(block: false, alt: "12")[$12$] is equal to #math.equation(block: false, alt: "17 .")[$17 .$]

The product of #math.equation(block: false, alt: "3")[$3$] and #math.equation(block: false, alt: "9")[$9$] is equal to #math.equation(block: false, alt: "27 .")[$27 .$]

#solutionbox[
3 ⋅ 9 = 27
]

The product of #math.equation(block: false, alt: "6")[$6$] and #math.equation(block: false, alt: "8")[$8$] is equal to #math.equation(block: false, alt: "48 .")[$48 .$]

The quotient of #math.equation(block: false, alt: "54")[$54$] and #math.equation(block: false, alt: "6")[$6$] is equal to #math.equation(block: false, alt: "9 .")[$9 .$]

#solutionbox[
54 ÷ 6 = 9
]

The quotient of #math.equation(block: false, alt: "42")[$42$] and #math.equation(block: false, alt: "7")[$7$] is equal to #math.equation(block: false, alt: "6 .")[$6 .$]

Twice the difference of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "10")[$10$] gives #math.equation(block: false, alt: "52 .")[$52 .$]

#solutionbox[
2(#emph[n] − 10) = 52
]

Twice the difference of #math.equation(block: false, alt: "m")[$m$] and #math.equation(block: false, alt: "14")[$14$] gives #math.equation(block: false, alt: "64 .")[$64 .$]

The sum of three times #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "10")[$10$] is #math.equation(block: false, alt: "100 .")[$100 .$]

#solutionbox[
3#emph[y] + 10 = 100
]

The sum of eight times #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "4")[$4$] is #math.equation(block: false, alt: "68 .")[$68 .$]

#strong[Translate to an Equation and Solve]

In the following exercises, translate the given sentence into an algebraic equation and then solve it.

Five more than #math.equation(block: false, alt: "p")[$p$] is equal to #math.equation(block: false, alt: "21 .")[$21 .$]

#solutionbox[
#emph[p] + 5 = 21; #emph[p] = 16
]

Nine more than #math.equation(block: false, alt: "q")[$q$] is equal to #math.equation(block: false, alt: "40 .")[$40 .$]

The sum of #math.equation(block: false, alt: "r")[$r$] and #math.equation(block: false, alt: "18")[$18$] is #math.equation(block: false, alt: "73 .")[$73 .$]

#solutionbox[
#emph[r] + 18 = 73; #emph[r] = 55
]

The sum of #math.equation(block: false, alt: "s")[$s$] and #math.equation(block: false, alt: "13")[$13$] is #math.equation(block: false, alt: "68 .")[$68 .$]

The difference of #math.equation(block: false, alt: "d")[$d$] and #math.equation(block: false, alt: "30")[$30$] is equal to #math.equation(block: false, alt: "52 .")[$52 .$]

#solutionbox[
#emph[d] − 30 = 52; #emph[d] = 82
]

The difference of #math.equation(block: false, alt: "c")[$c$] and #math.equation(block: false, alt: "25")[$25$] is equal to #math.equation(block: false, alt: "75 .")[$75 .$]

#math.equation(block: false, alt: "12")[$12$] less than #math.equation(block: false, alt: "u")[$u$] is #math.equation(block: false, alt: "89 .")[$89 .$]

#solutionbox[
#emph[u] − 12 = 89; #emph[u] = 101
]

#math.equation(block: false, alt: "19")[$19$] less than #math.equation(block: false, alt: "w")[$w$] is #math.equation(block: false, alt: "56 .")[$56 .$]

#math.equation(block: false, alt: "325")[$325$] less than #math.equation(block: false, alt: "c")[$c$] gives #math.equation(block: false, alt: "799 .")[$799 .$]

#solutionbox[
#emph[c] − 325 = 799; #emph[c] = 1124
]

#math.equation(block: false, alt: "299")[$299$] less than #math.equation(block: false, alt: "d")[$d$] gives #math.equation(block: false, alt: "850 .")[$850 .$]


==== Everyday Math

#strong[Insurance] Vince’s car insurance has a #math.equation(block: false, alt: "$500")[$"$500"$] deductible. Find the amount the insurance company will pay, #math.equation(block: false, alt: "p ,")[$p ,$] for an #math.equation(block: false, alt: "$1800")[$"$1800"$] claim by solving the equation #math.equation(block: false, alt: "500 plus p equals 1800 .")[$500 + p = 1800 .$]

#solutionbox[
\$1300
]

#strong[Insurance] Marta’s homeowner’s insurance policy has a #math.equation(block: false, alt: "$750")[$"$750"$] deductible. The insurance company paid #math.equation(block: false, alt: "$5800")[$"$5800"$] to repair damages caused by a storm. Find the total cost of the storm damage, #math.equation(block: false, alt: "d ,")[$d ,$] by solving the equation #math.equation(block: false, alt: "d minus 750 equals 5800 .")[$d − 750 = 5800 .$]

#strong[Sale purchase] Arthur bought a suit that was on sale for #math.equation(block: false, alt: "$120")[$"$120"$] off. He paid #math.equation(block: false, alt: "$340")[$"$340"$] for the suit. Find the original price, #math.equation(block: false, alt: "p ,")[$p ,$] of the suit by solving the equation #math.equation(block: false, alt: "p minus 120 equals 340 .")[$p − 120 = 340 .$]

#solutionbox[
\$460
]

#strong[Sale purchase] Rita bought a sofa that was on sale for #math.equation(block: false, alt: "$1299 .")[$"$1299" .$] She paid a total of #math.equation(block: false, alt: "$1409 ,")[$"$1409" ,$] including sales tax. Find the amount of the sales tax, #math.equation(block: false, alt: "t ,")[$t ,$] by solving the equation #math.equation(block: false, alt: "1299 plus t equals 1409 .")[$1299 + t = 1409 .$]


==== Writing Exercises

Is #math.equation(block: false, alt: "x equals 1")[$x = 1$] a solution to the equation #math.equation(block: false, alt: "8 x minus 2 equals 16 minus 6 x ?")[$8 x − 2 = 16 − 6 x ?$] How do you know?

Write the equation #math.equation(block: false, alt: "y minus 5 equals 21")[$y − 5 = 21$] in words. Then make up a word problem for this equation.


==== Self Check

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

#figure(figph[A self-assessment chart for math skills, including determining solutions, modeling and solving equations using addition/subtraction properties, and translating word phrases to algebraic equations.], alt: "A self-assessment chart for math skills, including determining solutions, modeling and solving equations using addition/subtraction properties, and translating word phrases to algebraic equations.", caption: none)

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