Modeling Linear Equations Word Problems- Free Worksheet with Solutions

What You'll Find Here

Below is a complete guide to modeling linear equations from word problems. It includes a free worksheet with 10 practice problems and full solutions. No fluff, no motivational garbage—just the math you need.

Why Word Problems Exist (And Why They're Not the Enemy)

Word problems force you to do two things at once: translate English into math and solve the resulting equation. Most students fail because they skip the translation step and try to solve before they have an equation.

The process is straightforward:

• Read the problem twice
• Identify what you're solving for (assign a variable)
• Find the relationship between quantities
• Write the equation
• Solve and check your answer

That's it. Stop overcomplicating it.

Key Translation Phrases to Memorize

These phrases appear constantly. Learn them or lose marks—your choice.

Common Problem Types

1. Number Problems

Find a number when you know how it relates to other numbers. Example: "Five more than a number is 12" becomes x + 5 = 12.

2. Age Problems

These involve present ages, past ages, or future ages. The key is that the age difference stays constant. If Dad is 30 years older than Tim, that difference never changes.

3. Distance-Rate-Time Problems

Use the formula distance = rate × time. If two objects move toward or away from each other, add or subtract their distances accordingly.

4. Mixture Problems

Usually involve combining solutions with different concentrations. Set up an equation based on the amount of pure substance before and after mixing.

5. Cost-Word Problems

Items have prices, you have a total budget, and you need to find how many of each item you can buy. Multiply items by their costs, then sum to the total.

How to Model Linear Equations: Step-by-Step

Let's work through a real example:

"Sarah has twice as many apples as Tom. Together they have 45 apples. How many does each have?"

Step 1: Identify the variable. Let x = number of apples Tom has.

Step 2: Express the other quantity. Sarah has 2x apples.

Step 3: Write the equation from the total. x + 2x = 45.

Step 4: Solve. 3x = 45, so x = 15.

Step 5: Answer the question. Tom has 15 apples, Sarah has 30.

Step 6: Check. 15 + 30 = 45 ✓ and 30 is twice 15 ✓

Common Mistakes That Cost You Points

• Forgetting to define the variable — if you don't say what x represents, you lose credit
• Setting up the equation wrong — "three less than a number" is x − 3, not 3 − x
• Not checking your answer — always plug your solution back into the original problem
• Solving for the wrong variable — read carefully: are you finding the number of items or the cost?

Free Worksheet: 10 Practice Problems

Solve each problem. Show your work. Answers are below.

1. A number increased by 7 equals 22. Find the number.
2. Three times a number minus 5 equals 16. What is the number?
3. Lisa is 4 years younger than her brother. Their ages sum to 26. How old is each?
4. Two numbers differ by 10. The larger number is 3 times the smaller. Find both numbers.
5. A taxi charges $3 base fare plus $2 per mile. A 17-mile ride costs how much?
6. You buy notebooks at $4 each and pens at $2 each, spending $24 total. You bought 2 more notebooks than pens. How many of each did you buy?
7. A train travels 60 mph faster than a car. The train covers 300 miles in the same time the car covers 120 miles. Find the car's speed.
8. Mixing 10% acid solution with 30% acid solution to get 20 liters of 18% solution. How much of each do you need?
9. A rectangle's length is 3 times its width. The perimeter is 80 cm. Find the dimensions.
10. Two angles are supplementary. One angle is 20° less than 3 times the other. Find both angles.

Solutions

1. x + 7 = 22 → x = 15
2. 3x − 5 = 16 → x = 7
3. Let x = brother's age. Lisa = x − 4. Equation: x + (x − 4) = 26 → x = 15. Brother is 15, Lisa is 11.
4. Let x = smaller number. Larger = x + 10 = 3x. Equation: x + 10 = 3x → x = 5. Numbers are 5 and 15.
5. Cost = 3 + 2(17) = $37
6. Let p = pens, n = notebooks. n = p + 2. Equation: 4(p + 2) + 2p = 24 → p = 4. You bought 4 pens and 6 notebooks.
7. Let car speed = x mph. Train = x + 60. Equation: 300/(x + 60) = 120/x → x = 40 mph (car). Train = 100 mph.
8. Let x = liters of 10% solution, y = liters of 30% solution. x + y = 20 and 0.10x + 0.30y = 3.6. Solution: x = 12 liters, y = 8 liters.
9. Let width = w. Length = 3w. Perimeter: 2(3w + w) = 80 → w = 10 cm, length = 30 cm.
10. Let smaller angle = x. Larger = 3x − 20. Equation: x + (3x − 20) = 180 → x = 50°. Angles are 50° and 130°.

Quick Reference: Equation Templates

When you're stuck, use these patterns:

• Sum problems: x + (x + a) = total
• Difference problems: x − (x − a) = difference
• Product problems: x × n = product
• Percentage problems: part = (percent/100) × whole
• Distance problems: d₁ + d₂ = total distance, or d₁ = d₂ when time is equal

Final Advice

Practice the translation step until it's automatic. Read a word problem, close your eyes, and say the equation out loud before you write anything. If you can do that, you can solve any linear equation word problem they throw at you.