1.7Problem Solving with Equations
Algebra becomes most powerful when you use it to solve real problems. This chapter teaches you to read a word problem, define a variable, write an equation, solve it, and check that your answer makes sense in context.
Why This Matters
Translating real-world situations into equations is the core skill of applied mathematics. Engineers, scientists, and business analysts do this every day — and it's the foundation of every word problem you'll see on the SAT and ACT.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do you translate a real-world situation into an algebraic equation — and how do you know when your answer is correct?
Lesson Overview
Every word problem is a story that hides an equation. The key skill is translation: converting English phrases into algebraic expressions and equations. Once you have an equation, you solve it using the techniques from Chapters 01–06. The final — and often skipped — step is to check your answer in the original problem, not just in your equation, to make sure it makes sense in context.
Worked Examples
Number Problem: Five more than three times a number is 26. Find the number.
Let x = the number
3x + 5 = 26
Subtract 5: 3x = 21
Divide by 3: x = 7
Check: 3(7)+5=26 ✓
Consecutive Integers: The sum of three consecutive integers is 48. Find the integers.
Let x, x+1, x+2 be the integers
x+(x+1)+(x+2)=48
3x+3=48 → 3x=45 → x=15
Check: 15+16+17=48 ✓
Geometry: A rectangle has perimeter 56 cm. The length is 4 cm more than the width. Find the dimensions.
Let w = width, l = w+4
2(w+4)+2w=56
4w+8=56 → 4w=48 → w=12
l=16
Check: 2(16)+2(12)=56 ✓
Angles: Two angles are supplementary. One is 30° more than twice the other. Find both angles.
Let x = smaller angle, 2x+30 = larger
x+(2x+30)=180
3x+30=180 → 3x=150 → x=50
Larger: 2(50)+30=130
Check: 50+130=180 ✓
Age Problem: Maria is 3 times as old as her brother. In 5 years she will be twice his age. How old is Maria now?
Let b = brother's age, Maria = 3b
In 5 years: 3b+5=2(b+5)
3b+5=2b+10 → b=5
Maria = 3(5) = 15
Check: 15+5=20; 5+5=10; 20=2(10) ✓
Guided Practice
Guided Practice Video: Problem Solving with Equations
Watch the guided practice walkthrough for problem solving with equations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Seven less than four times a number is 21. Find the number.
Hint: Let x = the number. Write: 4x − 7 = 21.
The sum of three consecutive integers is 63. Find the integers.
Hint: Let x, x+1, x+2 be the integers. Write their sum equal to 63.
A rectangle's length is twice its width. The perimeter is 48 m. Find the dimensions.
Hint: Let w = width, l = 2w. Use P = 2l + 2w = 48.
Two angles are complementary. One is 18° more than the other. Find both angles.
Hint: Let x = smaller angle. Complementary means x + (x+18) = 90.
One number is 5 more than another. Their sum is 41. Find both numbers.
Hint: Let x = smaller number, x+5 = larger. Write: x + (x+5) = 41.
Key Vocabulary
Define the Variable
Writing a clear statement of what the variable represents before writing the equation.
Example: Let x = the number of apples.
Consecutive Integers
Integers that follow each other in order, differing by 1.
Example: n, n+1, n+2 (e.g., 5, 6, 7)
Consecutive Even/Odd Integers
Even or odd integers in order, differing by 2.
Example: n, n+2, n+4 (e.g., 4, 6, 8)
Supplementary Angles
Two angles whose measures sum to 180°.
Example: x + (2x + 30) = 180
Practice Questions
Interactive Practice — 5 Questions
The sum of a number and 15 is 42. Which equation models this?
Three consecutive integers have a sum of 48. What is the smallest?
A rectangle has perimeter 56 cm. Its length is 4 more than its width. What is the width?
Maria is 3 times as old as her brother. In 5 years she will be twice his age. How old is Maria now?
Two trains leave the same station in opposite directions. One travels at 60 mph, the other at 80 mph. After how many hours are they 420 miles apart?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Twice a number increased by 7 is 31. Find the number.
The sum of four consecutive integers is 82. Find the integers.
A rectangle has perimeter 80 ft. The length is 5 ft more than three times the width. Find the dimensions.
One number is 8 less than another. Their sum is 52. Find both numbers.
A gym membership costs $25 to join plus $15 per month. After how many months will the total cost be $115?
Common Mistakes
Setting up the equation before defining what x represents.
Always start with 'Let x = …' to clearly define your variable before writing the equation.
Translating 'less than' in the wrong order — e.g., 'five less than a number' written as 5 − x.
'Five less than x' means x − 5. The phrase reverses the order.
Stopping at the value of x when the problem asks for something else.
Re-read the question after solving. If x is the smaller number, compute the larger one from x.
Using the wrong gap for consecutive integers — e.g., using +1 for consecutive even integers.
Consecutive integers: x, x+1. Consecutive even or odd integers: x, x+2.
Math Tips
Always write "Let x = ___" before writing your equation — it forces you to be precise about what you are solving for.
"Less than" reverses order: "3 less than x" is x − 3, not 3 − x.
Draw a diagram for geometry problems — it helps you see which sides or angles are related.
After solving, re-read the question. Sometimes the problem asks for something other than x.