Two-Step Equations

Solving Two-Step Equations

A two-step equation requires two inverse operations to solve.

Strategy: Undo operations in reverse order (undo addition/subtraction first, then multiplication/division).

Example 1

$3x + 5 = 20$

Step 1: Subtract 5 from both sides: $3x = 15$ Step 2: Divide both sides by 3: $x = 5$ ✅

Check: $3(5) + 5 = 15 + 5 = 20$ ✓

Example 2

$\frac{x}{4} - 3 = 7$

Step 1: Add 3 to both sides: $\frac{x}{4} = 10$ Step 2: Multiply both sides by 4: $x = 40$ ✅

Example 3 (Negative coefficient)

$-2x + 8 = 14$

Step 1: Subtract 8: $-2x = 6$ Step 2: Divide by $-2$: $x = -3$ ✅

Equations with Fractions

$\frac{2}{3}x - 4 = 8$

Step 1: Add 4: $\frac{2}{3}x = 12$ Step 2: Multiply by $\frac{3}{2}$: $x = 12 \times \frac{3}{2} = 18$ ✅

Two-Step Inequalities

Same process, but flip the inequality sign when multiplying or dividing by a negative.

$-3x + 9 > 21$ $-3x > 12$ $x < -4$ (flipped because we divided by $-3$)

Word Problems

"A number doubled, then decreased by 7, equals 15. Find the number." $2n - 7 = 15 \implies 2n = 22 \implies n = 11$

Golden rule: Whatever you do to one side, you MUST do to the other side.