Substitution Method for Linear Equations

The Substitution Method

An algebraic approach to find exact solutions
when graphical methods aren't practical!

Three Steps to Success

1

Express One Variable

Find one variable (y) in terms of the other (x)

2

Substitute

Substitute into the other equation and solve

3

Back-Substitute

Find the other variable using the first result

Example 4: Basic Substitution

Solve the system:

7x - 15y = 2

x + 2y = 3

Step 1: Express x in terms of y

From: x + 2y = 3

x = 3 - 2y

Step 2: Substitute & Solve

7(3 - 2y) - 15y = 2

21 - 14y - 15y = 2

-29y = -19

y = 19/29

Step 3: Find x

x = 3 - 2(19/29)

x = 87/29 - 38/29

x = 49/29

Solution

x = 49/29, y = 19/29

Example 5: Aftab's Age Problem

"Seven years ago, I was seven times as old as you."
"Three years from now, I'll be three times as old as you."

Setting up equations

Let s = Aftab's age, t = daughter's age

s - 7 = 7(t - 7) → s - 7t + 42 = 0

s + 3 = 3(t + 3) → s - 3t = 6

Solving

From s - 3t = 6:

s = 3t + 6

Substitute into s - 7t + 42 = 0:

(3t + 6) - 7t + 42 = 0

-4t + 48 = 0 → t = 12

Then: s = 3(12) + 6

s = 42

Answer

Aftab: 42 years, Daughter: 12 years

Example 6: Infinite Solutions

2 pencils + 3 erasers = ₹9
4 pencils + 6 erasers = ₹18

2x + 3y = 9 → x = (9 - 3y)/2

Substitute into 4x + 6y = 18:

4(9 - 3y)/2 + 6y = 18

18 - 6y + 6y = 18

18 = 18 ✓ (Always true!)

Result

Infinitely Many Solutions

Example 7: No Solution

Will these railway tracks cross?

x + 2y - 4 = 0 → x = 4 - 2y

Substitute into 2x + 4y - 12 = 0:

2(4 - 2y) + 4y - 12 = 0

8 - 4y + 4y - 12 = 0

-4 = 0 ✗ (False!)

Result

No Solution - Parallel!