Solving Equations

Notes

Solving step by step

• Use inverse operations on both sides to get the variable alone. Undo addition and subtraction first, then multiplication and division.
• Simplify each side first: distribute and combine like terms. Clear fractions by multiplying every term by the least common denominator.
• With variables on both sides, move them to one side. A false result like $2 = 9$ means no solution; a true result like $4 = 4$ means infinitely many solutions.
• Check by substituting your answer into the original equation.

Formulas, proportions and percents

• Solve literal equations for a chosen variable the same way: $\displaystyle A = \frac{1}{2}bh \Rightarrow h = \frac{2A}{b}$.
• Solve proportions with cross products: $\displaystyle \frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$.
• Percent change $\displaystyle = \frac{\text{new} - \text{original}}{\text{original}}$.

Absolute value

$|ax + b| = c$ with $c > 0$ splits into $ax + b = c$ or $ax + b = -c$. If $c < 0$ there is no solution.