2.6: Solving Equations Using the Subtraction and Addition Properties of Equality (Part 2)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 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 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.
Translate the sentence into an algebraic equation: The sum of \(6\) and \(9\) is \(15\).
Solution
The word is tells us the equal sign goes between \(9\) and \(15\).
| Locate the “equals” word(s). | The sum of 6 and 9 is 15 |
| Write the = sign. | The sum of 6 and 9 = 15 |
| Translate the words to the left of the equals word into an algebraic expression. | 6 + 9 = _____ |
| Translate the words to the right of the equals word into an algebraic expression. | 6 + 9 = 15 |
Translate the sentence into an algebraic equation: The sum of \(7\) and \(6\) gives \(13\).
- Answer
-
\(7+6=13\)
Translate the sentence into an algebraic equation: The sum of \(8\) and \(6\) is \(14\).
- Answer
-
\(8+6=14\)
Translate the sentence into an algebraic equation: The product of \(8\) and \(7\) is \(56\).
Solution
The location of the word is tells us that the equal sign goes between \(7\) and \(56\).
| Locate the “equals” word(s). | The product of 8 and 7 is 56 |
| Write the = sign. | The product of 8 and 7 = 56 |
| Translate the words to the left of the equals word into an algebraic expression. | 8 • 7 = _____ |
| Translate the words to the right of the equals word into an algebraic expression. | 8 • 7 = 56 |
Translate the sentence into an algebraic equation: The product of \(6\) and \(9\) is \(54\).
- Answer
-
\(6\cdot 9 = 54\)
Translate the sentence into an algebraic equation: The product of \(21\) and \(3\) gives \(63\).
- Answer
-
\(21\cdot 3 = 63\)
Translate the sentence into an algebraic equation: Twice the difference of \(x\) and \(3\) gives \(18\).
Solution
| Locate the “equals” word(s). |  |
| Recognize the key words: twice; difference of … and … | Twice means two times. |
| Translate. |  |
Translate the given sentence into an algebraic equation: Twice the difference of \(x\) and \(5\) gives \(30\).
- Answer
-
\(2(x-5)=30\)
Translate the given sentence into an algebraic equation: Twice the difference of \(y\) and \(4\) gives \(16\).
- Answer
-
\(2(y-4)=16\)
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.
Translate and solve: Three more than \(x\) is equal to \(47\).
Solution
| Three more than x is equal to 47. | |
| Translate. | \(x + 3 = 47\) |
| Subtract 3 from both sides of the equation. | \(x + 3 \textcolor{red}{-3} = 47 \textcolor{red}{-3}\) |
| Simplify | \(x = 44\) |
| We can check. Let x = 44. | \(44 + 3 \stackrel{?}{=} 47\) |
| \(47 = 47 \; \checkmark\) |
So \(x = 44\) is the solution.
Translate and solve: Seven more than \(x\) is equal to \(37\).
- Answer
-
\(x + 7 = 37; x = 30\)
Translate and solve: Eleven more than \(y\) is equal to \(28\).
- Answer
-
\(y + 11 = 28; y = 17\)
Translate and solve: The difference of \(y\) and \(14\) is \(18\).
Solution
| The difference of y and 14 is 18. | |
| Translate. | \(y - 14 = 18\) |
| Add 14 to both sides. | \(y - 14 \textcolor{red}{+14} = 18 \textcolor{red}{+14}\) |
| Simplify. | \(y = 32\) |
| We can check. Let y = 32. | \(32 - 14 \stackrel{?}{=} 18\) |
| \(18 = 18 \; \checkmark\) |
So \(y = 32\) is the solution.
Translate and solve: The difference of \(z\) and \(17\) is equal to \(37\).
- Answer
-
\(z - 17 = 37; z = 54\)
Translate and solve: The difference of \(x\) and \(19\) is equal to \(45\).
- Answer
-
\(x - 19 = 45; x = 64\)
Access Additional Online Resources
Key Concepts
- 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.
- Subtraction Property of Equality
- For any numbers \(a\), \(b\), and \(c\),
- 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.
- Addition Property of Equality
- For any numbers \(a\), \(b\), and \(c\),
- 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.
Glossary
- solution of an equation
-
A 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.
Practice Makes Perfect
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.
- x + 13 = 21
- x = 8
- x = 34
- y + 18 = 25
- y = 7
- y = 43
- m − 4 = 13
- m = 9
- m = 17
- n − 9 = 6
- n = 3
- n = 15
- 3p + 6 = 15
- p = 3
- p = 7
- 8q + 4 = 20
- q = 2
- q = 3
- 18d − 9 = 27
- d = 1
- d = 2
- 24 f − 12 = 60
- f = 2
- f = 3
- 8u − 4 = 4u + 40
- u = 3
- u = 11
- 7v − 3 = 4v + 36
- v = 3
- v = 11
- 20h − 5 = 15h + 35
- h = 6
- h = 8
- 18k − 3 = 12k + 33
- k = 1
- k = 6
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.
Solve Equations using the Subtraction Property of Equality
In the following exercises, solve each equation using the subtraction property of equality.
- a + 2 = 18
- b + 5 = 13
- p + 18 = 23
- q + 14 = 31
- r + 76 = 100
- s + 62 = 95
- 16 = x + 9
- 17 = y + 6
- 93 = p + 24
- 116 = q + 79
- 465 = d + 398
- 932 = c + 641
Solve Equations using the Addition Property of Equality
In the following exercises, solve each equation using the addition property of equality.
- y − 3 = 19
- x − 4 = 12
- u − 6 = 24
- v − 7 = 35
- f − 55 = 123
- g − 39 = 117
- 19 = n − 13
- 18 = m − 15
- 10 = p − 38
- 18 = q − 72
- 268 = y − 199
- 204 = z − 149
Translate Word Phrase to Algebraic Equations
In the following exercises, translate the given sentence into an algebraic equation.
- The sum of 8 and 9 is equal to 17.
- The sum of 7 and 9 is equal to 16.
- The difference of 23 and 19 is equal to 4.
- The difference of 29 and 12 is equal to 17.
- The product of 3 and 9 is equal to 27.
- The product of 6 and 8 is equal to 48.
- The quotient of 54 and 6 is equal to 9.
- The quotient of 42 and 7 is equal to 6.
- Twice the difference of n and 10 gives 52.
- Twice the difference of m and 14 gives 64.
- The sum of three times y and 10 is 100.
- The sum of eight times x and 4 is 68.
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 p is equal to 21.
- Nine more than q is equal to 40.
- The sum of r and 18 is 73.
- The sum of s and 13 is 68.
- The difference of d and 30 is equal to 52.
- The difference of c and 25 is equal to 75.
- 12 less than u is 89.
- 19 less than w is 56.
- 325 less than c gives 799.
- 299 less than d gives 850.
Everyday Math
- Insurance Vince’s car insurance has a $500 deductible. Find the amount the insurance company will pay, p, for an $1800 claim by solving the equation 500 + p = 1800.
- Insurance Marta’s homeowner’s insurance policy has a $750 deductible. The insurance company paid $5800 to repair damages caused by a storm. Find the total cost of the storm damage, d, by solving the equation d − 750 = 5800.
- Sale purchase Arthur bought a suit that was on sale for $120 off. He paid $340 for the suit. Find the original price, p, of the suit by solving the equation p − 120 = 340.
- Sale purchase Rita bought a sofa that was on sale for $1299. She paid a total of $1409, including sales tax. Find the amount of the sales tax, t, by solving the equation 1299 + t = 1409.
Writing Exercises
- Is x = 1 a solution to the equation 8x − 2 = 16 − 6x ? How do you know?
- Write the equation y − 5 = 21 in words. Then make up a word problem for this equation.
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Contributors and Attributions
- Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.