What is a Two-Step Equation?
A two-step equation is an algebraic equation that requires exactly two inverse operations to solve for the unknown variable x.
The fundamental rule of algebra is the Golden Rule of Balance:
Whatever inverse operation you do to one side of the equal sign, you must do to the other side to keep the equation balanced.
The Secret Strategy: Reverse Order of Operations (SADMEP)
When calculating arithmetic expressions, you follow PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
When solving for an unknown variable, you do the exact opposite—you unwrap the variable in reverse order:
- S / A: Undo Subtraction and Addition first.
- D / M: Undo Division and Multiplication second.
Worked Example 1: Standard Two-Step Equation
Problem:
Solve for x:
4x + 9 = 33
Step 1: Undo the addition of 9
The opposite of + 9 is - 9. Subtract 9 from both sides:
Step 2: Undo the multiplication of 4
The coefficient 4 is multiplied by x. The inverse operation is division. Divide both sides by 4:
Step 3: Check your answer
Substitute x = 6 back into the original equation:
The equation is balanced, so x = 6 is guaranteed correct!
Worked Example 2: Equations with Fractions and Negatives
Many students get nervous when they see fractions or negative numbers, but the two steps never change!
Problem:
Solve for x:
x-3 - 7 = -2
Step 1: Undo subtraction first
The opposite of - 7 is + 7. Add 7 to both sides:
Step 2: Undo the division
\left(\frac{x}{-3}\right) \times (-3) = 5 \times (-3)
x = -15
\frac{-15}{-3} - 7 = 5 - 7 = -2 \quad \checkmark
Here, the entire expression x + 5 is grouped on top of the fraction bar (which acts like parentheses!).
In this specific case, multiply by the denominator first:
- Multiply both sides by 4:
- Subtract 5 from both sides:
Quick Reference Summary
| Equation Type | Example | Step 1 (S/A) | Step 2 (D/M) | Solution |
|---|---|---|---|---|
| Standard | 3x + 7 = 22 | Subtract 7: 3x = 15 | Divide by 3 | x = 5 |
| Negative Coefficient | -2x - 5 = 11 | Add 5: -2x = 16 | Divide by -2 | x = -8 |
| Fraction Division | x5 + 3 = 10 | Subtract 3: x5 = 7 | Multiply by 5 | x = 35 |
| Grouped Fraction | x - 42 = 8 | Multiply by 2: x - 4 = 16 | Add 4 | x = 20 |
Don't Forget Negative Signs on Coefficients!
If you have -4x = 20, dividing both sides by 4 leaves -x = 5. Always divide by the full coefficient including its sign (-4), so x = -5.