Linear Equations in One Variable

x = 0

Step 1: We have 8x + 4 = 3(x - 1) + 7. First expand the bracket on RHS. Step 2: RHS = 3x - 3 + 7 = 3x + 4. Step 3: Equation becomes 8x + 4 = 3x + 4. Step 4: Transpose: 8x - 3x = 4 - 4 → 5x = 0 → x = 0. Step 5: Verify — LHS = 8(0) + 4 = 4; RHS = 3(0-1) + 7 = -3 + 7 = 4. LHS = RHS ✓ Common mistake: Not expanding the bracket correctly — students may forget to multiply 3 × (-1) = -3.

x = 40

Step 1: We have x = (4/5)(x + 10). Multiply both sides by 5 to remove the fraction. Step 2: 5x = 4(x + 10). Step 3: Expand: 5x = 4x + 40. Step 4: Transpose: 5x - 4x = 40 → x = 40. Step 5: Verify — RHS = (4/5)(40 + 10) = (4/5)(50) = 40 = LHS ✓ Common mistake: Students multiply only the numerator by 5 and forget to multiply (x+10), leading to incorrect expansion.

x = 10

Step 1: We have (2x/3) + 1 = (7x/15) + 3. LCM of 3 and 15 is 15. Step 2: Multiply every term by 15: 15×(2x/3) + 15×1 = 15×(7x/15) + 15×3. Step 3: Simplify: 10x + 15 = 7x + 45. Step 4: Transpose: 10x - 7x = 45 - 15 → 3x = 30 → x = 10. Step 5: Verify — LHS = (20/3) + 1 = 23/3; RHS = (70/15) + 3 = 14/3 + 9/3 = 23/3. LHS = RHS ✓ Common mistake: Using LCM incorrectly or multiplying only some terms by 15.

x = -1

Step 1: LCM of 3 and 6 is 6. Multiply every term by 6. Step 2: 6×(6x+1)/3 + 6×1 = 6×(x-3)/6 → 2(6x+1) + 6 = (x-3). Step 3: Expand: 12x + 2 + 6 = x - 3 → 12x + 8 = x - 3. Step 4: Transpose: 12x - x = -3 - 8 → 11x = -11 → x = -1. Step 5: Verify — LHS = (6(-1)+1)/3 + 1 = (-5/3) + 1 = -2/3; RHS = (-1-3)/6 = -4/6 = -2/3. LHS = RHS ✓ Common mistake: Not expanding 2(6x+1) properly — missing 2×1 = 2.