Cross Multiplication — Solving Linear Equations

Use when an equation is in the form:

a/b = c/d (or reducible to this form — a fraction on each side)

Cross multiplication converts: a/b = c/d → a × d = b × c

If p/q = r/s, then ps = qr

This works because multiplying both sides by qs gives: p·s = r·q

Example 1: Solve (x+1)/3 = (x−2)/2

Step 1: Cross multiply → 2(x+1) = 3(x−2)
Step 2: Expand → 2x + 2 = 3x − 6
Step 3: Rearrange → 2 + 6 = 3x − 2x
Step 4: Solve → x = 8

Verify: LHS = (8+1)/3 = 3 | RHS = (8−2)/2 = 3 ✓

Example 2: Solve (2x−3)/5 = (x+1)/4

Step 1: Cross multiply → 4(2x−3) = 5(x+1)
Step 2: Expand → 8x − 12 = 5x + 5
Step 3: Rearrange → 8x − 5x = 5 + 12
Step 4: Solve → 3x = 17 → x = 17/3

Example 3: Solve 3/(x+2) = 5/(2x−1)

Step 1: Cross multiply → 3(2x−1) = 5(x+2)
Step 2: Expand → 6x − 3 = 5x + 10
Step 3: Solve → x = 13

"A number divided by 5 gives the same result as 3 less than the number divided by 4."

Let the number = x: x/5 = (x−3)/4 Cross multiply: 4x = 5(x−3) → 4x = 5x − 15 → x = 15

1. (x+3)/(x−3) = 5/3 → solve for x
2. If (2x+1)/5 = (3x−2)/7, find x
3. A fraction = 3/5. Numerator + denominator = 32. Find the fraction.

Solution to Q3: Let numerator = n, denominator = d. Given: n/d = 3/5 → 5n = 3d. Also: n + d = 32 → d = 32 − n → 5n = 3(32−n) = 96 − 3n → 8n = 96 → n = 12, d = 20. Fraction = 12/20 = 3/5 ✓

1. Solve: (y+2)/3 = (y−1)/2
2. Solve: 4/(x+1) = 2/(x−3)
3. Can you cross-multiply x + 3/4 = 7? Why or why not?
4. A ratio of two numbers is 5:7. Their sum is 144. Find the numbers.
5. Stretch: Solve (2x+1)/(x+3) = (3x−2)/(2x+1) using cross multiplication.