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Algebra I

Word Problems with Linear Equations

Lesson

Most “real life” problems become easy once you get past the words and onto the page. Use the same playbook every time.

The four-step playbook

  1. Find the unknown and name it xx.
  2. Write an equation that captures the relationship.
  3. Solve the equation.
  4. Check by plugging the answer back into the original sentence.

Worked example 1 — flat fee + per-unit cost

A gym charges $25 to join plus $8 per visit. If your total bill is $89, how many visits did you make?

25+8x=8925 + 8x = 89
8x=64  ⟹  x=88x = 64 \implies x = 8

Worked example 2 — relating two quantities

Two numbers sum to 25, and one is 7 more than the other. Find the smaller.

x+(x+7)=25x + (x + 7) = 25
2x=18  ⟹  x=92x = 18 \implies x = 9

Worked example 3 — mix of two items

A farm has chickens (2 legs) and cows (4 legs). 30 animals, 80 legs. How many chickens?

2c+4(30−c)=802c + 4(30 - c) = 80
2c+120−4c=80  ⟹  c=202c + 120 - 4c = 80 \implies c = 20

Translation cheat

  • “Per” signals multiplication.
  • “Plus a flat fee” signals a constant added.
  • “Sum” / “total” → addition.
  • “Difference” / “less than” → subtraction.
  • “Of” with a percent or fraction → multiplication.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

5 more than twice a number is 13. Find the number.\text{5 more than twice a number is 13. Find the number.}

Problem 2

A number plus 6 is 19. Find the number.\text{A number plus 6 is 19. Find the number.}

Problem 3

Three times a number is 24. Find the number.\text{Three times a number is 24. Find the number.}

Problem 4

9 less than a number is −2. Find the number.\text{9 less than a number is } -2. \text{ Find the number.}

Practice

Standard problems matching the lesson.

Problem 5

Gym: 25 + 8x = 89. Visits?

Problem 6

Taxi: 4 + 2x = 16. Miles?

Problem 7

Phone: 20 + 5x = 50. GB?

Problem 8

Rectangle perimeter 30, length 12. Width?

Problem 9

Two numbers sum 25, one is 7 more. Smaller?

Problem 10

Mike = Sara + 4. Mike + Sara = 28. Sara?

Problem 11

Three consecutive integers sum 36. Smallest?

Problem 12

2x + 5 = 17. x?

Problem 13

3x + 4 = 22. x?

Problem 14

Equilateral triangle perimeter 24. Side?

Problem 15

Car rental: 50 + 0.25m = 75. Miles?

Problem 16

8 - x = 5. Slices eaten?

Problem 17

Lee = 3 * son. Lee + son = 60. Son?

Problem 18

2c + 4(30-c) = 80. Chickens?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

45000 + 1500y = 60000. Years?

Problem 20

Two numbers differ by 8, sum 24. Larger?

Problem 21

30 + 0.10m = 40 + 0.05m. Minutes?

Problem 22

3h + 5(12-h) = 42. Hot dogs?

Problem 23

Triangle sides x, x+2, x+4 sum 30. x?

Problem 24

L = W + 3. Perimeter 38. Width?

Problem 25

1000 + 25m = 1500. Months?

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