Training Linear Equations and Inequalities Word Problems with Equations
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Word Problems with Equations

22 min Linear Equations and Inequalities

Translating Word Problems

Word problems are where algebra meets the real world. Translating a sentence like "the sum of a number and seven is twenty-three" into the equation x + 7 = 23 is the first and most important step — once the equation is set up correctly, the solving techniques you have already learned take over.

This lesson presents a variety of word problems that require setting up and solving linear equations. You will encounter age problems, consecutive-integer problems, distance-rate-time problems, and mixture problems.

The key to success is a methodical approach: define your variable, write the equation, solve, and then check your answer against the original problem statement.

Key Phrases
PhraseSymbol
"is," "equals," "gives"$=$
"more than," "increased by"$+$
"less than," "decreased by"$-$
"times," "of," "product"$\times$
"at least"$\ge$
"at most," "no more than"$\le$
Example 1

Tom is 5 years older than twice Sara's age. Tom is 37. How old is Sara?

Let $s$ = Sara's age. $\;2s + 5 = 37 \;\Rightarrow\; 2s = 32 \;\Rightarrow\; s = 16$.

Example 2

Three consecutive even integers sum to 72.

$n + (n+2) + (n+4) = 72 \;\Rightarrow\; 3n + 6 = 72 \;\Rightarrow\; n = 22$.

Integers: $22, 24, 26$.

Example 3

Phone plan: $\$45$/mo + $\$0.10$/text. Budget $\le \$70$. Max texts?

$45 + 0.10t \le 70 \;\Rightarrow\; 0.10t \le 25 \;\Rightarrow\; t \le 250$.

Example 4

Two angles are supplementary. One is $30°$ more than twice the other. Find both.

$x + (2x + 30) = 180 \;\Rightarrow\; 3x = 150 \;\Rightarrow\; x = 50$. Angles: $50°$ and $130°$.

Example 5

A rectangle's length is 3 more than twice its width. Perimeter is 48. Dimensions?

$w + (2w + 3) + w + (2w + 3) = 48 \;\Rightarrow\; 6w + 6 = 48 \;\Rightarrow\; w = 7$.

Width $= 7$, Length $= 17$.

Practice Problems

1. A number doubled and increased by 7 equals 25. Find it.
2. Three consecutive integers sum to 108.
3. Gym: $\$50$ signup $+ \$30$/mo. When does total exceed $\$260$?
4. Two complementary angles; one is $15°$ less than twice the other.
5. A car rents for $\$35$/day $+ \$0.20$/mi. Budget $\le \$100$/day. Max miles?
6. Father is 4 times as old as son. In 10 years, he'll be twice as old. Ages now?
7. Sum of two numbers is 50. One is 8 more than the other.
8. Perimeter of equilateral triangle is 45 cm. Side length?
9. Score $\ge 80$ avg on 4 tests. First 3 scores: 75, 82, 88. Min 4th score?
10. Taxi: $\$2.50$ base $+ \$0.40$/quarter-mile. Trip costs $\$8.50$. How far?
Show Answer Key

1. $2x + 7 = 25$; $x = 9$

2. $35, 36, 37$

3. $50 + 30m > 260$; after month 7

4. $x + (2x - 15) = 90$; $35°$ and $55°$

5. $35 + 0.20m \le 100$; $m \le 325$ mi

6. Son $= 5$, Father $= 20$

7. $21$ and $29$

8. $15$ cm

9. $\dfrac{75+82+88+x}{4} \ge 80$; $x \ge 75$

10. $15$ quarter-miles $= 3.75$ miles

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