📐 Equations I: Solving Strategies & Techniques
Master every equation type on the SAT — from basic linear to absolute value, radicals, and quadratics
Part 1 — Fundamentals
1.1 What Is an Equation?
An equation is a mathematical statement that two expressions are equal. It always contains an equals sign = and usually contains one or more variables (unknowns) whose values we want to find.
1.2 What Is a Solution?
A solution is any value of the variable that makes the equation a true statement. When you substitute the solution back in, both sides are equal.
- An equation may have one solution, no solution, or infinitely many solutions.
- Example: $3x + 5 = 20$ → $x = 5$ because $3(5)+5 = 20$ ✓
1.3 Equivalent Equations
Two equations are equivalent if they have the exact same solution set. Our goal when solving is to transform an equation into simpler and simpler equivalent forms until we reach $x = \text{number}$.
Both equations have the same solution: $x = 3$.
1.4 The Golden Rule of Equations
Part 2 — Tools for Isolating Variables
Every equation-solving technique boils down to one idea: isolate the variable. Here are the tools you'll use.
| Tool | What It Does | Example |
|---|---|---|
| Addition / Subtraction Property | Add or subtract the same value from both sides to remove a term | $x + 5 = 12 \;\Rightarrow\; x = 12 - 5 = 7$ |
| Multiplication / Division Property | Multiply or divide both sides by the same non-zero value to remove a coefficient | $3x = 15 \;\Rightarrow\; x = \frac{15}{3} = 5$ |
| Inverse Operations | Addition ↔ Subtraction; Multiplication ↔ Division; Squaring ↔ Square Root | Use the inverse to "undo" what's been done to $x$ |
| Reverse PEMDAS | Undo operations in reverse order: undo +/− first, then ×/÷, then exponents | $2x + 3 = 11$: undo +3 first, then undo ×2 |
| Negative Coefficients | $-x = 7$ means $x = -7$; multiply or divide both sides by $-1$ | $-x = 7 \;\Rightarrow\; x = -7$ |
| Fractional Coefficients | Multiply both sides by the reciprocal | $\frac{2}{3}x = 8 \;\Rightarrow\; x = 8 \times \frac{3}{2} = 12$ |
| Cross Multiplication | If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$ | $\frac{x}{4} = \frac{3}{5} \;\Rightarrow\; 5x = 12 \;\Rightarrow\; x = \frac{12}{5}$ |
| Clearing Fractions (LCD) | Multiply every term by the LCD to eliminate all fractions | $\frac{x}{2} + \frac{x}{3} = 5$; LCD = 6 → $3x + 2x = 30$ |
| Clearing Decimals | Multiply by a power of 10 | $0.3x + 1.5 = 2.7$ → multiply by 10: $3x + 15 = 27$ |
| Combine Like Terms | Simplify each side first | $3x + 2x + 5 = 20 \;\Rightarrow\; 5x + 5 = 20$ |
| Distribute | Expand parentheses before solving | $3(x + 4) = 21 \;\Rightarrow\; 3x + 12 = 21$ |