Linear Equations
A linear equation contains a variable raised to the first power and forms a straight line when graphed. The goal is to isolate the variable by performing the same operation on both sides of the equation. Linear equations appear in forms like x + 5 = 12 or 3x - 7 = 14, where the variable has no exponents or radicals.
Why it matters
Linear equations model countless real-world relationships where one quantity changes at a constant rate relative to another. A phone plan charging $30 plus $0.10 per text follows the linear equation C = 30 + 0.1t, where C represents total cost and t represents texts sent. Businesses use linear equations to calculate break-even points — if a company's costs are $500 plus $2 per item and revenue is $7 per item, they break even when 500 + 2x = 7x, solving to x = 100 items. Linear equations also appear in distance-rate-time problems, where driving 60 miles per hour for t hours covers d = 60t miles. Mastering linear equations prepares students for systems of equations in CCSS.8.EE and advanced algebra topics in CCSS.HSA.REI, forming the foundation for understanding linear functions and graphing.
How to solve linear equations
Linear equations — how to
- Collect x-terms on one side, constants on the other.
- Do the same operation to both sides (add, subtract, multiply, divide).
- Divide by the coefficient of x to isolate x.
Example: 3x + 7 = 22 → 3x = 15 → x = 5.
Worked examples
x + 5 = 12
Answer: x = 7
- Subtract 5 from both sides → x = 12 − 5 — To isolate x, undo the addition.
- Calculate → x = 7 — 12 − 5 = 7.
- Verify → 7 + 5 = 12 ✓ — Substitution confirms the solution.
2x − 3 = 15
Answer: x = 9
- Add 3 to both sides → 2x = 18 — Isolate the x term by removing the constant.
- Divide both sides by 2 → x = 9 — 18 ÷ 2 = 9.
- Verify → 2(9) − 3 = 15 ✓ — Substitution confirms the solution.
8x − 18 = 5x + 3
Answer: x = 7
- Subtract 5x from both sides → 3x − 18 = 3 — Collect all x terms on one side.
- Add 18 to both sides → 3x = 21 — Move constants to the other side.
- Divide both sides by 3 → x = 7 — 21 ÷ 3 = 7.
- Verify → LHS = RHS = 38 ✓ — Both sides equal the same value.
Common mistakes
- Applying operations to only one side of the equation, such as solving x + 5 = 12 by writing x = 12 + 5 = 17 instead of x = 12 - 5 = 7.
- Incorrectly combining like terms when variables appear on both sides, such as solving 5x + 3 = 2x + 15 by writing 3x + 3 = 15 instead of properly moving terms to get 3x = 12.
- Sign errors when moving terms across the equals sign, such as solving 2x - 7 = 9 by writing 2x = 9 - 7 = 2 instead of 2x = 9 + 7 = 16.