0Can you solve this in your head?
2 pens and 3 notebooks cost ₹80. 1 pen and 2 notebooks cost ₹50. What does one pen cost?
Feels tricky? By the end of this lesson, you'll crack it in under a minute.
Step 1: Turn the story into equations
Let pen = x and notebook = y. Then:
- 2x + 3y = 80
- x + 2y = 50
That's a pair of linear equations in two variables. Any word problem can be turned into one like this.
Step 2: Pick your weapon (Substitution)
From the second equation: x = 50 − 2y.
Put it in the first: 2(50 − 2y) + 3y = 80
100 − 4y + 3y = 80, so y = 20.
Then x = 50 − 40 = 10.
A pen costs ₹10 and a notebook costs ₹20.
Check: 2(10) + 3(20) = 80 ✓ and 10 + 2(20) = 50 ✓. Always verify, since it takes 10 seconds and saves marks.
Step 3: The faster weapon (Elimination)
Solve: 2x + y = 7 and x − y = 2.
Notice that +y and −y cancel when you add the equations.
Add them: 3x = 9, so x = 3.
Put x = 3 in x − y = 2, so y = 1.
Solution: x = 3, y = 1.
Trick: when a variable has opposite signs, add. When it has the same sign, subtract.
Step 4: The secret exam question, "How many solutions?"
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, compare the ratios:
| Condition | Lines look like | Solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Cross each other | Exactly one |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Same line | Infinitely many |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel lines | None |
Memory trick: Cross = one, Same = many, Parallel = none.
Quick test: 2x + 3y = 9 and 4x + 6y = 18. Every ratio equals 1/2, so they're the same line and have infinitely many solutions.
Your turn (practice question)
Solve: x + y = 10 and x − y = 4.
Message me your answer, and I'll check it.
Exam tips
- Always define your variables first ("Let the number of pens be x").
- Write the final answer in a sentence for word problems.
- Substitute back to check. Examiners love verified answers.
Want to make word problems your strongest topic? Message me on UrbanPro to book a free demo class. I break every chapter into simple steps so students feel confident, not scared.
