Grade 8 Linear Equations: Special Cases

Worked examples

Real problems from the practice worksheets, solved one step at a time.

1. Example 1

   How many solutions does 4x − 5 = 3x − 6 + x have: one, none, or infinitely many?

   1. Combine like terms: 3x − 6 + x = 4x − 6, so 4x − 5 = 4x − 6.
   2. Subtract 4x from both sides: −5 = −6.
   3. −5 = −6 is never true, so no value of x works. The equation has no solution.

   Answer: no solution

2. Example 2

   How many solutions does 6(x − 7) = 7x + 1 have: one, none, or infinitely many?

   1. Distribute the 6: 6(x − 7) = 6x − 42, so 6x − 42 = 7x + 1.
   2. The x-terms are different (6x on the left, 7x on the right), so they don't cancel: exactly one value of x makes the sides equal. The equation has one solution.

   Answer: one solution

3. Example 3

   Solve: 8x + 2 = 24 − 3x

   1. Add 3x to both sides: 11x + 2 = 24.
   2. Subtract 2 from both sides: 11x = 24 − 2 = 22.
   3. Divide both sides by 11: x = 22 ÷ 11 = 2.
   4. Check: 8 × 2 + 2 = 18 and 24 − 3 × 2 = 18 ✓ So x = 2.

   Answer: 2