Algebra I to AP Precalculus · Module 2 of 14
Solving Linear Equations
Isolating a variable through one-step, multi-step, literal, and absolute value equations.
One-Step and Two-Step Equations
- Use inverse operations to isolate the variable, undoing addition/subtraction before multiplication/division (reverse order of operations).
- Whatever operation you apply to one side of the equation, apply to the other side too, to keep it balanced.
- Check your solution by substituting it back into the original equation.
Solve 2x + 5 = 17.
Solution. Subtract 5 from both sides: 2x = 12. Divide both sides by 2: x = 6. Check: 2(6) + 5 = 17 ✓.
Multi-Step Equations (Variables on Both Sides)
- Distribute and combine like terms on each side separately first.
- Move all variable terms to one side (by adding/subtracting) and all constants to the other.
- Divide by the coefficient of the variable to finish.
Solve 3(x − 2) = 2x + 4.
Solution. Distribute: 3x − 6 = 2x + 4. Subtract 2x: x − 6 = 4. Add 6: x = 10. Check: 3(8) = 24 = 2(10) + 4 ✓.
Common mistake: Distribute before combining or moving terms — combining across the equals sign too early causes sign errors.
Literal Equations & Formulas
- A literal equation has more than one variable; solve for one variable "in terms of" the others using the same inverse-operation steps.
- Treat every other variable as if it were a constant number while you isolate the one you want.
Solve for h: A = (1/2)bh (area of a triangle).
Solution. Multiply both sides by 2: 2A = bh. Divide both sides by b: h = 2A/b.
Solving Absolute Value Equations
- |x| = a (a > 0) means x is a units from 0, so it splits into two equations: x = a or x = −a.
- Isolate the absolute value expression completely before splitting into the two cases.
- If a is negative, |expression| = a has no solution, since absolute value can never be negative.
Solve |2x − 3| = 7.
Solution. Split into two cases: 2x − 3 = 7 → x = 5, or 2x − 3 = −7 → x = −2. Solutions: x = 5 or x = −2.
Common mistake: Isolate the absolute value bars completely (undo anything added outside them) before splitting into cases.
Check yourself
2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.
1. Solve: 4x − 7 = 13.
2. Solve: 5(x + 1) = 3x + 11.