Solving linear equations
A linear equation in one variable can be written ax + b = c. The goal is to get x alone by doing the same thing to both sides. Work in a fixed order:
- Remove brackets by distributing.
- Collect like terms on each side.
- Move the x-terms to one side and numbers to the other.
- Divide by the coefficient of x.
Worked example
Solve 3(x − 2) + 5 = 2x + 9.
- Distribute: 3x − 6 + 5 = 2x + 9
- Combine: 3x − 1 = 2x + 9
- Subtract 2x: x − 1 = 9
- Add 1: x = 10
Check by substituting: left side 3(8) + 5 = 29; right side 20 + 9 = 29.
Lines and slope
The slope of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁): rise over run. Through (1, 2) and (5, 10) the slope is 8/4 = 2.
Three useful forms:
- Slope-intercept: y = mx + b, where b is the y-intercept.
- Point-slope: y − y₁ = m(x − x₁).
- Standard: Ax + By = C.
Parallel lines have equal slopes. Perpendicular lines have slopes that multiply to −1 (negative reciprocals). To find an x-intercept, set y = 0; for the y-intercept, set x = 0.
Inequalities
Solve a linear inequality like an equation with one extra rule: multiplying or dividing by a negative number flips the inequality sign.
Solve −2x + 3 > 11. Subtract 3: −2x > 8. Divide by −2 and flip: x < −4. The solution is every number below −4, written (−∞, −4).
A compound inequality such as 1 ≤ 2x − 3 < 7 is solved in one go by doing each step to all three parts: add 3 to get 4 ≤ 2x < 10, divide by 2 to get 2 ≤ x < 5.
Absolute value
|x| is the distance of x from zero. So:
- |x − 3| = 5 means x is 5 units from 3, giving x = 8 or x = −2.
- |x| < 4 means the distance is less than 4, so −4 < x < 4 (one connected interval).
- |x| > 4 means x < −4 or x > 4 (two separate pieces).
Common mistakes
- Forgetting to flip the sign when dividing by a negative.
- Giving only one answer to an absolute-value equation.
- Not distributing a negative sign across a whole bracket.