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Solve Equations Using the Subtraction and Addition Properties of Equality

Solve equations using the Subtraction and Addition Properties of Equality

Solve Equations Using the Subtraction and Addition Properties of Equality

We began our work solving equations in previous chapters. It has been a while since we have seen an equation, so we will review some of the key concepts before we go any further.

We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.

In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.

Example

Try it.

Determine whether $y=\frac{3}{4}$ is a solution for $4y+3=8y.$

Solution



Multiply.
Add.

Since $y=\frac{3}{4}$ results in a true equation, $\frac{3}{4}$ is a solution to the equation $4y+3=8y.$

We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.

When you add or subtract the same quantity from both sides of an equation, you still have equality.

We introduced the Subtraction Property of Equality earlier by modeling equations with envelopes and counters. models the equation $x+3=8.$

The goal is to isolate the variable on one side of the equation. So we ‘took away’ $3$ from both sides of the equation and found the solution $x=5.$

Example

Try it.

Solve: $m+4=-5.$

Solution


Subtract 4 from each side to "undo" the addition.
Simplify.
Check:
Substitute $m=-9$.

		The solution to $m-4=-5$ is $m=-1$.

Condensed — the full section is in OpenStax Prealgebra 2e.

Solve Equations That Need to Be Simplified

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

Example

Try it.

Solve: $3x-7-2x-4=1.$

Solution

The left side of the equation has an expression that we should simplify before trying to isolate the variable.


Rearrange the terms, using the Commutative Property of Addition.
Combine like terms.
Add 11 to both sides to isolate $x$.
Simplify.
Check. Substitute $x=12$ into the original equation.

The solution checks.

Example

Try it.

Solve: $3(n-4)-2n=-3.$

Solution

The left side of the equation has an expression that we should simplify.


Distribute on the left.
Use the Commutative Property to rearrange terms.
Combine like terms.
Isolate n using the Addition Property of Equality.
Simplify.
Check. Substitute $n=9$ into the original equation. The solution checks.

Example

Try it.

Solve: $2(3k-1)-5k=-2-7.$

Solution

Both sides of the equation have expressions that we should simplify before we isolate the variable.


Distribute on the left, subtract on the right.
Use the Commutative Property of Addition.
Combine like terms.
Undo subtraction by using the Addition Property of Equality.
Simplify.
Check. Let $k=-7.$ The solution checks.

Translate an Equation and Solve

In previous chapters, we translated word sentences into equations. The first step is to look for the word (or words) that translate(s) to the equal sign. reminds us of some of the words that translate to the equal sign.

Equals (=)
is	is equal to	is the same as	the result is	gives	was	will be

Let’s review the steps we used to translate a sentence into an equation.

Now we are ready to try an example.

Example

Try it.

Translate and solve: five more than $x$ is equal to $26.$

Solution

Translate.
Subtract 5 from both sides.
Simplify.
Check: Is $26$ five more than $21$? The solution checks.

Example

Try it.

Translate and solve: The difference of $5p$ and $4p$ is $23.$

Solution

Translate.
Simplify.
Check:
The solution checks.

Translate and Solve Applications

In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will be using equations to solve application problems. We’ll start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for.

Example

Try it.

The Robles family has two dogs, Buster and Chandler. Together, they weigh $71$ pounds.

Chandler weighs $28$ pounds. How much does Buster weigh?

Solution

Read the problem carefully.
Identify what you are asked to find, and choose a variable to represent it.	How much does Buster weigh? Let $b=$ Buster's weight
Write a sentence that gives the information to find it.	Buster's weight plus Chandler's weight equals 71 pounds.
We will restate the problem, and then include the given information.	Buster's weight plus 28 equals 71.
Translate the sentence into an equation, using the variable $b$.
Solve the equation using good algebraic techniques.
Check the answer in the problem and make sure it makes sense.
Is 43 pounds a reasonable weight for a dog? Yes. Does Buster's weight plus Chandler's weight equal 71 pounds?
$43+28\overset{?}{=}71$
$71=71\ ✓$
Write a complete sentence that answers the question, "How much does Buster weigh?"	Buster weighs 43 pounds

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Determine whether a number is a solution to an equation.
1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
3. 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.
Subtraction and Addition Properties of Equality
- Subtraction Property of Equality
  For all real numbers a, b, and c,
  if a = b then $a-c=b-c$.
- Addition Property of Equality
  For all real numbers a, b, and c,
  if a = b then $a+c=b+c$.
Translate a word sentence to an algebraic equation.
1. Locate the “equals” word(s). Translate to an equal sign.
2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
3. Translate the words to the right of the “equals” word(s) into an algebraic expression.
Problem-solving strategy
1. Read the problem. Make sure you understand all the words and ideas.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

Solve Equations Using the Subtraction and Addition Properties of Equality

Solve Equations Using the Subtraction and Addition Properties of Equality

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

Try it.

Is $y=\frac{1}{3}$ a solution of $4y+2=10y?$

Solution

yes

Try it.

Is $x=\frac{3}{4}$ a solution of $5x+3=9x?$

Try it.

Is $u=-\frac{1}{2}$ a solution of $8u-1=6u?$

Solution

Try it.

Is $v=-\frac{1}{3}$ a solution of $9v-2=3v?$

In the following exercises, solve each equation.

Try it.

$x+7=12$

Solution

x = 5

Try it.

$y+5=-6$

Try it.

$b+\frac{1}{4}=\frac{3}{4}$

Solution

$b=\frac{1}{2}$

Try it.

$a+\frac{2}{5}=\frac{4}{5}$

Try it.

$p+2.4=-9.3$

Solution

p = −11.7

Try it.

$m+7.9=11.6$

Try it.

$a-3=7$

Solution

a = 10

Try it.

$m-8=-20$

Try it.

$x-\frac{1}{3}=2$

Solution

$x=\frac{7}{3}$

Try it.

$x-\frac{1}{5}=4$

Try it.

$y-3.8=10$

Solution

y = 13.8

Try it.

$y-7.2=5$

Try it.

$x-15=-42$

Solution

x = −27

Try it.

$z+5.2=-8.5$

Try it.

$q+\frac{3}{4}=\frac{1}{2}$

Solution

$q=-\frac{1}{4}$

Try it.

$p-\frac{2}{5}=\frac{2}{3}$

Try it.

$y-\frac{3}{4}=\frac{3}{5}$

Solution

$y=\frac{27}{20}$

Solve Equations that Need to be Simplified

In the following exercises, solve each equation.

Try it.

$c+3-10=18$

Try it.

$m+6-8=15$

Solution

m = 17

Try it.

$9x+5-8x+14=20$

Try it.

$6x+8-5x+16=32$

Solution

x = 8

Try it.

$-6x-11+7x-5=-16$

Try it.

$-8n-17+9n-4=-41$

Solution

n = −20

Try it.

$3(y-5)-2y=-7$

Try it.

$4(y-2)-3y=-6$

Solution

y = 2

Try it.

$8(u+1.5)-7u=4.9$

Try it.

$5(w+2.2)-4w=9.3$

Solution

w = −1.7

Try it.

$-5(y-2)+6y=-7+4$

Try it.

$-8(x-1)+9x=-3+9$

Solution

x = −2

Try it.

$3(5n-1)-14n+9=1-2$

Try it.

$2(8m+3)-15m-4=3-5$

Solution

m = −4

Try it.

$-(j+2)+2j-1=5$

Try it.

$-(k+7)+2k+8=7$

Solution

k = 6

Try it.

$6a-5(a-2)+9=-11$

Try it.

$8c-7(c-3)+4=-16$

Solution

c = −41

Try it.

$8(4x+5)-5(6x)-x=53$

Try it.

$6(9y-1)-10(5y)-3y=22$

Solution

y = 28

Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve.

Try it.

Five more than $x$ is equal to $21.$

Try it.

The sum of $x$ and $-5$ is $33.$

Solution

x + (−5) = 33; x = 38

Try it.

Ten less than $m$ is $-14.$

Try it.

Three less than $y$ is $-19.$

Solution

y − 3 = −19; y = −16

Try it.

The sum of $y$ and $-3$ is $40.$

Try it.

Eight more than $p$ is equal to $52.$

Solution

p + 8 = 52; p = 44

Try it.

The difference of $9x$ and $8x$ is $17.$

Try it.

The difference of $5c$ and $4c$ is $60.$

Solution

5c − 4c = 60; c = 60

Try it.

The difference of $n$ and $\frac{1}{6}$ is $\frac{1}{2}.$

Try it.

The difference of $f$ and $\frac{1}{3}$ is $\frac{1}{12}.$

Solution

$f-\frac{1}{3}=\frac{1}{12};\ f=\frac{5}{12}$

Try it.

The sum of $-4n$ and $5n$ is $-32.$

Try it.

The sum of $-9m$ and $10m$ is $-25.$

Solution

−9m + 10m = −25; m = −25

Translate and Solve Applications

In the following exercises, translate into an equation and solve.

Try it.

Pilar drove from home to school and then to her aunt’s house, a total of $18$ miles. The distance from Pilar’s house to school is $7$ miles. What is the distance from school to her aunt’s house?

Try it.

Jeff read a total of $54$ pages in his English and Psychology textbooks. He read $41$ pages in his English textbook. How many pages did he read in his Psychology textbook?

Solution

Let p equal the number of pages read in the Psychology book. 41 + p = 54. Jeff read 13 pages in his Psychology book.

Try it.

Pablo’s father is $3$ years older than his mother. Pablo’s mother is $42$ years old. How old is his father?

Try it.

Eva’s daughter is $5$ years younger than her son. Eva’s son is $12$ years old. How old is her daughter?

Solution

Let d equal the daughter’s age. d = 12 − 5. Eva’s daughter’s age is 7 years old.

Try it.

Allie weighs $8$ pounds less than her twin sister Lorrie. Allie weighs $124$ pounds. How much does Lorrie weigh?

Try it.

For a family birthday dinner, Celeste bought a turkey that weighed $5$ pounds less than the one she bought for Thanksgiving. The birthday dinner turkey weighed $16$ pounds. How much did the Thanksgiving turkey weigh?

Solution

21 pounds

Try it.

The nurse reported that Tricia’s daughter had gained $4.2$ pounds since her last checkup and now weighs $31.6$ pounds. How much did Tricia’s daughter weigh at her last checkup?

Try it.

Connor’s temperature was $0.7$ degrees higher this morning than it had been last night. His temperature this morning was $101.2$ degrees. What was his temperature last night?

Solution

100.5 degrees

Try it.

Melissa’s math book cost $\text{\$22.85}$ less than her art book cost. Her math book cost $\text{\$93.75}.$ How much did her art book cost?

Try it.

Ron’s paycheck this week was $\text{\$17.43}$ less than his paycheck last week. His paycheck this week was $\text{\$103.76}.$ How much was Ron’s paycheck last week?

Solution

$121.19

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Solve: $n-12=16.$
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

$28$
Translate into algebra ‘five less than $x\text{.’}$
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

$x-5$
Is $x=2$ a solution to $5x-3=7?$
If you missed this problem, review .

Fi àwọn àgbèwọlé hàn

yes
Determine whether $y=\frac{3}{4}$ is a solution for $4y+3=8y.$

Fi àwọn àgbèwọlé hàn

Multiply.
Add.
Since $y=\frac{3}{4}$ results in a true equation, $\frac{3}{4}$ is a solution to the equation $4y+3=8y.$
Is $y=\frac{2}{3}$ a solution for $9y+2=6y?$

Fi àwọn àgbèwọlé hàn

no
Is $y=\frac{2}{5}$ a solution for $5y-3=10y?$

Fi àwọn àgbèwọlé hàn

no
Solve: $x-11=-3.$

Fi àwọn àgbèwọlé hàn

To isolate $x,$ we undo the addition of $11$ by using the Subtraction Property of Equality.
We "undo" the subtraction of 11 by adding 11 to each side.
Simplify.
Check:
Substitute $x=8$.
Since $x=8$ makes $x-11=-3$ a true statement, we know that it is a solution to the equation.
Solve: $x+9=-7.$

Fi àwọn àgbèwọlé hàn

x = −16
Solve: $x+16=-4.$

Fi àwọn àgbèwọlé hàn

x = −20
Solve: $m+4=-5.$

Fi àwọn àgbèwọlé hàn

Subtract 4 from each side to "undo" the addition.
Simplify.
Check:
Substitute $m=-9$.
The solution to $m-4=-5$ is $m=-1$.
Solve: $n-6=-7.$

Fi àwọn àgbèwọlé hàn

n = −1
Solve: $x-5=-9.$

Fi àwọn àgbèwọlé hàn

x = −4
Solve: $n-\frac{3}{8}=\frac{1}{2}.$

Fi àwọn àgbèwọlé hàn

Use the Addition Property of Equality.
Find the LCD to add the fractions on the right.
Simplify
Check:
Subtract.
Simplify.
The solution checks.
Solve: $p-\frac{1}{3}=\frac{5}{6}.$

Fi àwọn àgbèwọlé hàn

$p=\frac{7}{6}$
Solve: $q-\frac{1}{2}=\frac{1}{6}.$

Fi àwọn àgbèwọlé hàn

$q=\frac{2}{3}$
Solve $a-3.7=4.3.$

Fi àwọn àgbèwọlé hàn

Use the Addition Property of Equality.
Add.
Check:
Substitute $a=8$.
Simplify.
The solution checks.
Solve: $b-2.8=3.6.$

Fi àwọn àgbèwọlé hàn

b = 6.4
Solve: $c-6.9=7.1.$

Fi àwọn àgbèwọlé hàn

c = 14
Solve: $3x-7-2x-4=1.$

Fi àwọn àgbèwọlé hàn

The left side of the equation has an expression that we should simplify before trying to isolate the variable.

Rearrange the terms, using the Commutative Property of Addition.
Combine like terms.
Add 11 to both sides to isolate $x$.
Simplify.
Check.
Substitute $x=12$ into the original equation.
The solution checks.
Solve: $8y-4-7y-7=4.$

Fi àwọn àgbèwọlé hàn

y = 15
Solve: $6z+5-5z-4=3.$

Fi àwọn àgbèwọlé hàn

z = 2

Solve: $3(n-4)-2n=-3.$

Fi àwọn àgbèwọlé hàn

The left side of the equation has an expression that we should simplify.


Distribute on the left.
Use the Commutative Property to rearrange terms.
Combine like terms.
Isolate n using the Addition Property of Equality.
Simplify.
Check. Substitute $n=9$ into the original equation. The solution checks.

Solve: $5(p-3)-4p=-10.$

Fi àwọn àgbèwọlé hàn

p = 5
Solve: $4(q+2)-3q=-8.$

Fi àwọn àgbèwọlé hàn

q = −16

Solve: $2(3k-1)-5k=-2-7.$

Fi àwọn àgbèwọlé hàn

Both sides of the equation have expressions that we should simplify before we isolate the variable.


Distribute on the left, subtract on the right.
Use the Commutative Property of Addition.
Combine like terms.
Undo subtraction by using the Addition Property of Equality.
Simplify.
Check. Let $k=-7.$ The solution checks.

Solve: $4(2h-3)-7h=-6-7.$

Fi àwọn àgbèwọlé hàn

h = −1
Solve: $2(5x+2)-9x=-2+7.$

Fi àwọn àgbèwọlé hàn

x = 1
Translate and solve: five more than $x$ is equal to $26.$

Fi àwọn àgbèwọlé hàn

Translate.
Subtract 5 from both sides.
Simplify.
Check:
Is $26$ five more than $21$?

The solution checks.
Translate and solve: Eleven more than $x$ is equal to $41.$

Fi àwọn àgbèwọlé hàn

x + 11 = 41; x = 30
Translate and solve: Twelve less than $y$ is equal to $51.$

Fi àwọn àgbèwọlé hàn

y − 12 = 51; y = 63
Translate and solve: The difference of $5p$ and $4p$ is $23.$

Fi àwọn àgbèwọlé hàn

Translate.
Simplify.
Check:

The solution checks.
Translate and solve: The difference of $4x$ and $3x$ is $14.$

Fi àwọn àgbèwọlé hàn

4x − 3x = 14; x = 14
Translate and solve: The difference of $7a$ and $6a$ is $-8.$

Fi àwọn àgbèwọlé hàn

7a − 6a = −8; a = −8

The Robles family has two dogs, Buster and Chandler. Together, they weigh $71$ pounds.

Chandler weighs $28$ pounds. How much does Buster weigh?

Fi àwọn àgbèwọlé hàn

Read the problem carefully.
Identify what you are asked to find, and choose a variable to represent it.	How much does Buster weigh? Let $b=$ Buster's weight
Write a sentence that gives the information to find it.	Buster's weight plus Chandler's weight equals 71 pounds.
We will restate the problem, and then include the given information.	Buster's weight plus 28 equals 71.
Translate the sentence into an equation, using the variable $b$.
Solve the equation using good algebraic techniques.
Check the answer in the problem and make sure it makes sense.
Is 43 pounds a reasonable weight for a dog? Yes. Does Buster's weight plus Chandler's weight equal 71 pounds?
$43+28\overset{?}{=}71$
$71=71\ ✓$
Write a complete sentence that answers the question, "How much does Buster weigh?"	Buster weighs 43 pounds

Translate into an algebraic equation and solve: The Pappas family has two cats, Zeus and Athena. Together, they weigh $13$ pounds. Zeus weighs $6$ pounds. How much does Athena weigh?

Fi àwọn àgbèwọlé hàn

a + 6 = 13; Athena weighs 7 pounds.
Translate into an algebraic equation and solve: Sam and Henry are roommates. Together, they have $68$ books. Sam has $26$ books. How many books does Henry have?

Fi àwọn àgbèwọlé hàn

26 + h = 68; Henry has 42 books.

Shayla paid $\text{\$24,575}$ for her new car. This was $\text{\$875}$ less than the sticker price. What was the sticker price of the car?

Fi àwọn àgbèwọlé hàn

What are you asked to find?	"What was the sticker price of the car?"
Assign a variable.	Let $s=$ the sticker price of the car.
Write a sentence that gives the information to find it.	$24,575 is $875 less than the sticker price $24,575 is $875 less than $s$
Translate into an equation.
Solve.
Check:
Is $875 less than $25,450 equal to $24,575?
$25,450-875\overset{?}{=}24,575$
$24,575=24,575\ ✓$
Write a sentence that answers the question.	The sticker price was $25,450.

Translate into an algebraic equation and solve: Eddie paid $\text{\$19,875}$ for his new car. This was $\text{\$1,025}$ less than the sticker price. What was the sticker price of the car?

Fi àwọn àgbèwọlé hàn

19,875 = s − 1025; the sticker price is $20,900.
Translate into an algebraic equation and solve: The admission price for the movies during the day is $\text{\$7.75}.$ This is $\text{\$3.25}$ less than the price at night. How much does the movie cost at night?

Fi àwọn àgbèwọlé hàn

7.75 = n − 3.25; the price at night is $11.00.
Is $y=\frac{1}{3}$ a solution of $4y+2=10y?$

Fi àwọn àgbèwọlé hàn

yes

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Solve Equations Using the Subtraction and Addition Properties of Equality

Solve equations using the Subtraction and Addition Properties of Equality
Solve equations that need to be simplified
Translate an equation and solve
Translate and solve applications
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.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Wárá

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Diẹ̀ nínú Arithmetic

Order of operations Fractions Percentages Powers and roots


Rearrange the terms, using the Commutative Property of Addition.
Combine like terms.
Add 11 to both sides to isolate \(x\).
Simplify.
Check. Substitute \(x=12\) into the original equation.

Read the problem carefully.
Identify what you are asked to find, and choose a variable to represent it.	How much does Buster weigh? Let \(b=\) Buster's weight
Write a sentence that gives the information to find it.	Buster's weight plus Chandler's weight equals 71 pounds.
We will restate the problem, and then include the given information.	Buster's weight plus 28 equals 71.
Translate the sentence into an equation, using the variable \(b\).
Solve the equation using good algebraic techniques.
Check the answer in the problem and make sure it makes sense.
Is 43 pounds a reasonable weight for a dog? Yes. Does Buster's weight plus Chandler's weight equal 71 pounds?
\(43+28\overset{?}{=}71\)
\(71=71\ ✓\)
Write a complete sentence that answers the question, "How much does Buster weigh?"	Buster weighs 43 pounds


We "undo" the subtraction of 11 by adding 11 to each side.
Simplify.
Check:
Substitute \(x=8\).


Use the Addition Property of Equality.
Find the LCD to add the fractions on the right.
Simplify
Check:

Subtract.
Simplify.
The solution checks.