Grade 8 Mathematics
A LINEAR EQUATION is like a balance scale - both sides must be equal. To solve, we isolate the variable by doing the SAME operation to BOTH sides.
Goal: Get the variable ALONE on one side!
STEP 1: SIMPLIFY - Distribute and combine like terms on each side
STEP 2: COLLECT - Move all variables to one side, constants to the other
STEP 3: ISOLATE - Get the variable term alone (undo addition/subtraction)
STEP 4: SOLVE - Divide to find the variable's value
Solve: 3(x + 4) - 2 = 16
SIMPLIFY: Distribute 3 to both terms in parentheses
3(x + 4) - 2 = 16 → 3x + 12 - 2 = 16 → 3x + 10 = 16
ISOLATE: Subtract 10 from both sides
3x + 10 - 10 = 16 - 10 → 3x = 6
SOLVE: Divide both sides by 3
3x / 3 = 6 / 3 → x = 2
CHECK: Substitute x = 2 back into the original equation
3(2 + 4) - 2 = 3(6) - 2 = 18 - 2 = 16 ✓
Solve: 5x + 7 = 2x + 22
COLLECT variables: Subtract 2x from BOTH sides to move all x's to one side
5x - 2x + 7 = 2x - 2x + 22 → 3x + 7 = 22
ISOLATE: Subtract 7 from both sides
3x + 7 - 7 = 22 - 7 → 3x = 15
SOLVE: Divide both sides by 3
3x / 3 = 15 / 3 → x = 5
CHECK: Left side: 5(5) + 7 = 32. Right side: 2(5) + 22 = 32. ✓
What if all the variables CANCEL OUT?
TRUE statement (like 5 = 5): INFINITE SOLUTIONS - Any number works!
FALSE statement (like 3 = 8): NO SOLUTION - No number works!
Solve: 4x + 5 = 4x + 10
Subtract 4x from both sides: 4x - 4x + 5 = 4x - 4x + 10
5 = 10 ← This is FALSE!
NO SOLUTION - No value of x can make 5 equal to 10!
Solve: 2(x + 3) = 2x + 6
Distribute: 2x + 6 = 2x + 6
Subtract 2x from both sides: 6 = 6
6 = 6 ← This is TRUE for ANY value of x!
INFINITE SOLUTIONS - Every number works!
1. Solve: 2x + 5 = 17
x =
2. Solve: 4(x - 3) = 20
x =
3. Solve: 6x - 4 = 3x + 11
x =
4. Solve: 5x + 2 = 5x + 7. What type of solution?
Solution type: