Write linear equations to solve word problems

Key notes:

What is a Linear Equation in a Word Problem?

A linear equation is an equation that models a real-world situation where variables have a power of 1.
It can be used to represent relationships between quantities in word problems.

General form:

ax+by=c

or

y=mx+b

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Steps to Write a Linear Equation from a Word Problem

• Understand the problem – Read carefully and identify what is being asked.
• Define the variables – Choose letters for the unknown quantities.
• Identify relationships – Find sentences that show how quantities are related.
• Translate to equation – Use algebraic symbols to represent the relationship.
• Solve the equation – Find the value(s) of the unknown(s).
• Interpret the answer – Write your answer in words.

💡 Think-Check: The equation should match the wording exactly.

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Worked Examples

Example 1 – Cost of items

A notebook costs ₹20 and a pen costs ₹10. If you buy 3 notebooks and 2 pens, the total cost is ₹80. Write a linear equation.

Step 1: Let
x= price of notebook
y = price of pen

But here prices are given, so instead let:
x= number of notebooks
y= number of pens

Step 2: Relation from problem:
Cost of notebooks =20x
Cost of pens =10y
Total cost = ₹80

Step 3: Write the equation:

20x+10y=80

✅ This is the linear equation representing the problem.

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Example 2 – Age problem

Ravi is 5 years older than twice his sister’s age. If Ravi is 19, write and solve the equation for his sister’s age.

Step 1: Let s = sister’s age.

Step 2: Relation:
Ravi’s age = 2s+5
Given Ravi’s age = 19.

Step 3: Equation:

2s+5=19

Step 4: Solve:
Subtract 5: 2s=14
Divide by 2: s=7

✅ Sister’s age is 7 years.

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Example 3 – Slope-intercept situation

A taxi charges ₹50 as a fixed fee plus ₹15 per kilometer. Write the equation for total cost C in terms of kilometers k.

Fixed fee → b=50
Rate per km → m=15

Equation:

C=15k+50

✅ This is in slope-intercept form y=mx+b

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Common Word Problem Clues for Equations

Words in problem	Mathematical meaning
total, sum, altogether	+
difference, less than, decreased by	−
product, times, of	×
per, rate, each	multiplication (slope)
is, gives, equals	=

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Practice Problems

Write a linear equation for each:

1. A fruit seller sells apples for ₹30 each and bananas for ₹10 each. If he sells a apples and b bananas for ₹300 total, write the equation.
2. A phone company charges ₹200 per month plus ₹2 for every minute of calls. Write the equation for total cost C in terms of minutes m
3. The length of a rectangle is 3 more than twice its width. Write the equation if the length is 15 cm.
4. The sum of two numbers is 50. One number is 4 less than twice the other. Write and solve the equation.

Learn with an example