11.3 Solving Linear Equations

11.3 Solving Linear Equations

Quick Answer: A linear equation in one variable has the variable to the first power only. Use inverse operations to isolate it: simplify both sides, collect like terms, get all variable terms on one side and constants on the other, then divide by the coefficient. Check by substituting your answer into the original equation.

Linear equations are the first listed mathematics competency on the McCann PERT outline. Placement items range from one-step “undo addition” problems to multi-step equations with parentheses and fractions. The method is always the same: maintain equality while peeling the equation until the variable stands alone.

Equality Balance Rule

Whatever you do to one side, do to the other:

• Add or subtract the same number on both sides
• Multiply or divide both sides by the same nonzero number

One-Step Equations

Worked Example 1 — Addition

Solve: x + 17 = 42

Subtract 17: x = 42 − 17 = 25.

Check: 25 + 17 = 42. ✓ Answer: x = 25.

Worked Example 2 — Multiplication

Solve: −6x = 48

Divide by −6: x = 48/(−6) = −8.

Check: −6(−8) = 48. ✓ Answer: x = −8.

Worked Example 3 — Division form

Solve: x/5 = −3

Multiply by 5: x = −15.

Check: −15/5 = −3. ✓

Two-Step Equations

Typical form: ax + b = c. Undo addition/subtraction first, then multiply/divide—or reverse if that is clearer—but stay consistent.

Worked Example 4

Solve: 3x − 7 = 14

Add 7: 3x = 21
Divide by 3: x = 7.

Check: 3(7) − 7 = 21 − 7 = 14. ✓ Answer: x = 7.

Worked Example 5 — Negative coefficient

Solve: −2x + 9 = 1

Subtract 9: −2x = −8
Divide by −2: x = 4.

Check: −2(4) + 9 = −8 + 9 = 1. ✓

Distractor path: Dividing only the −8 by −2 but “forgetting” a sign → x = −4 fails the check.

Variables on Both Sides

Worked Example 6

Solve: 5x + 3 = 2x + 18

Subtract 2x from both sides: 3x + 3 = 18
Subtract 3: 3x = 15
Divide by 3: x = 5.

Check: Left 5(5) + 3 = 28; right 2(5) + 18 = 28. ✓ Answer: x = 5.

Worked Example 7

Solve: 4 − 3x = 7x − 16

Add 3x: 4 = 10x − 16
Add 16: 20 = 10x
Divide by 10: x = 2.

Check: Left 4 − 3(2) = 4 − 6 = −2; right 7(2) − 16 = 14 − 16 = −2. ✓

Parentheses and the Distributive Property

a(b + c) = ab + ac. Distribute before combining like terms. A negative outside flips every sign inside.

Worked Example 8

Solve: 2(x + 4) = 3x − 1

Distribute: 2x + 8 = 3x − 1
Subtract 2x: 8 = x − 1
Add 1: x = 9.

Check: Left 2(9 + 4) = 2(13) = 26; right 3(9) − 1 = 27 − 1 = 26. ✓ Answer: x = 9.

Worked Example 9 — Negative distribution

Solve: 5 − 2(3x − 4) = 3x + 1

Distribute −2: 5 − 6x + 8 = 3x + 1
Combine left constants: 13 − 6x = 3x + 1
Add 6x: 13 = 9x + 1
Subtract 1: 12 = 9x
x = 12/9 = 4/3.

Check: Left 5 − 2(3·4/3 − 4) = 5 − 2(4 − 4) = 5 − 2(0) = 5
Right 3(4/3) + 1 = 4 + 1 = 5. ✓ Answer: x = 4/3.

Trap: Writing 5 − 2(3x − 4) as 5 − 6x − 4 (forgetting to flip the −4) leads to a wrong linear equation and a clean-looking but incorrect answer choice.

Fractional Coefficients

Clear fractions by multiplying every term by the least common denominator (LCD).

Worked Example 10

Solve: (1/2)x + 3 = (5/4)x − 1

LCD of 2 and 4 is 4. Multiply through by 4:
4·(1/2)x + 4·3 = 4·(5/4)x − 4·1
2x + 12 = 5x − 4
Subtract 2x: 12 = 3x − 4
Add 4: 16 = 3x
x = 16/3.

Check (optional decimal): 0.5(16/3) + 3 = 8/3 + 3 = 17/3
(5/4)(16/3) − 1 = (5·4)/3 − 1 = 20/3 − 3/3 = 17/3. ✓

Worked Example 11 — Simple fraction equation

Solve: x/3 − 2 = 4

Add 2: x/3 = 6
Multiply by 3: x = 18.

Or multiply original by 3 first: x − 6 = 12 → x = 18.

Infinite Solutions vs. No Solution (identity/contradiction)

After simplifying:

• If you get 0 = 0 or 5 = 5, every real number works (identity).
• If you get 0 = 7 or 3 = −1, no solution (contradiction).

Worked Example 12

2(x + 3) = 2x + 6 → 2x + 6 = 2x + 6 → 0 = 0 → infinitely many solutions.
2(x + 3) = 2x + 5 → 2x + 6 = 2x + 5 → 6 = 5 → no solution.

PERT-Style Multiple Choice with Error Distractors

PERT will not show your work. Options are engineered from common mistakes:

Worked Example 13 — Full PERT-style item

Solve: 4(x − 2) = 2x + 10

Distribute: 4x − 8 = 2x + 10
Subtract 2x: 2x − 8 = 10
Add 8: 2x = 18
x = 9.

Check: Left 4(9 − 2) = 28; right 2(9) + 10 = 28. ✓

Typical wrong options: x = 1 (if someone does 4x − 2 instead of 4x − 8), x = 3 (arithmetic slip on 18/2), x = −1 (sign flip when moving terms).

End-to-End Method Card

1. Simplify each side (distribute, combine like terms).
2. Move variable terms to one side; constants to the other.
3. Divide/multiply to isolate x.
4. Substitute back into the original equation.
5. If the check fails, re-examine distribution and sign moves first.

Linear equations dominate early PERT Math success. Combined with solid order of operations and careful evaluation, you can treat every solution as a verifiable number—not a guess.