Elimination Method for Linear Equations

The Elimination Method

Remove one variable by making coefficients equal,
then add or subtract equations!

Four Steps to Success

1

Make Coefficients Equal

Multiply equations to make one variable's coefficients equal

2

Eliminate Variable

Add or subtract equations to eliminate that variable

3

Solve

Solve the resulting equation in one variable

4

Back-Substitute

Find the other variable using the original equation

Example 8: Income & Savings Problem

Income ratio = 9:7, Expenditure ratio = 4:3
Both save ₹2000 per month

Let incomes be 9x and 7x, expenditures be 4y and 3y

9x - 4y = 2000

7x - 3y = 2000

Step 1: Make y coefficients equal

Multiply equation 1 by 3, equation 2 by 4:

27x - 12y = 6000

28x - 12y = 8000

Step 2: Eliminate y by subtraction

(28x - 12y) - (27x - 12y) = 8000 - 6000

x = 2000

Substitute x = 2000 into 9x - 4y = 2000:

9(2000) - 4y = 2000

18000 - 4y = 2000 → y = 4000

Monthly Incomes

₹18,000 and ₹14,000

Example 9: No Solution Case

Use elimination to solve:

2x + 3y = 8

4x + 6y = 7

Solving by Elimination

Multiply equation 1 by 2:

4x + 6y = 16

4x + 6y = 7

Subtract:

(4x + 6y) - (4x + 6y) = 16 - 7

0 = 9 ✗ (False!)

Result

No Solution - Inconsistent!

Example 10: Two-Digit Number

Number + Reversed = 66
Digits differ by 2

Let digits be x (tens) and y (units)
Number = 10x + y, Reversed = 10y + x

Setting up equations

From sum condition:

(10x + y) + (10y + x) = 66

11(x + y) = 66 → x + y = 6

Digits differ by 2:

x - y = 2 OR y - x = 2

Finding both numbers

Case 1: x - y = 2

x + y = 6, x - y = 2 → Add: 2x = 8 → x = 4, y = 2

Number = 42

Case 2: y - x = 2

x + y = 6, y - x = 2 → Add: 2y = 8 → y = 4, x = 2

Number = 24

Two Numbers Found

42 and 24