One- and Two-Step
Equations
Master the foundational skill of solving for unknown variables using inverse operations: the building block of all algebra.
ISEE Upper Level
What Is an Equation?
An equation is a mathematical statement that two expressions are equal. The equals sign \( = \) acts like a balance point: whatever is on the left must have the same value as what is on the right.
Key Vocabulary
Variable
A letter (like \( x \) or \( n \)) representing an unknown number you need to find.
Inverse Operation
The operation that "undoes" another: addition undoes subtraction, multiplication undoes division.
Isolate
To get the variable alone on one side of the equation by removing everything else.
Why Equations Matter
One- and two-step equations appear throughout the ISEE quantitative reasoning section. They are also the foundation for multi-step equations, inequalities, systems of equations, and word problems you will encounter later.
ISEE Test
Equations appear in roughly 15-20% of quantitative reasoning questions.
Building Block
Every advanced algebra topic depends on fluently solving basic equations.
The Golden Rule of Equations
One-Step vs Two-Step
| One-Step | Two-Step | |
|---|---|---|
| Operations to undo | 1 | 2 |
| Example | \( x + 7 = 15 \) | \( 3x + 7 = 22 \) |
| Strategy | Apply one inverse operation | Undo addition/subtraction first, then multiplication/division |
| Typical ISEE difficulty | Low | Medium |
How to Solve Step-by-Step
1
Identify the variable and the operations acting on it.
2
Undo addition or subtraction first by performing the inverse on both sides.
3
Undo multiplication or division next by performing the inverse on both sides.
4
Check your answer by substituting it back into the original equation.
Example 1: One-Step Equation
Example 2: One-Step with Multiplication
Example 3: Two-Step Equation
Example 4: Negatives and Fractions
1
Add 3 to both sides: \( \dfrac{x}{4} = 8 \)
2
Multiply both sides by 4: \( x = 32 \)
✓
Check: \( \dfrac{32}{4} - 3 = 8 - 3 = 5 \) ✓
Example 5: Variable on the Right Side
1
Subtract 5 from both sides: \( 12 = -2x \)
2
Divide both sides by \( -2 \): \( x = -6 \)
✓
Check: \( 5 - 2(-6) = 5 + 12 = 17 \) ✓
Common Mistakes
✕ Wrong Operation Order
Dividing before subtracting in a two-step equation. Always undo addition/subtraction first.
✕ One-Sided Operation
Performing an operation on only one side. Remember: both sides must stay balanced.
✕ Sign Errors
Forgetting to flip the sign when dividing by a negative. If \( -2x = 12 \), then \( x = -6 \), not \( 6 \).
Summary Reference
| Equation Type | Example | Steps | Key Inverse |
|---|---|---|---|
| Addition | \( x + 5 = 12 \) | 1 | Subtract |
| Subtraction | \( x - 9 = 14 \) | 1 | Add |
| Multiplication | \( 4x = 36 \) | 1 | Divide |
| Division | \( \frac{x}{3} = 7 \) | 1 | Multiply |
| Two-Step | \( 3x + 7 = 22 \) | 2 | Subtract, then Divide |
| Two-Step (fraction) | \( \frac{x}{4} - 3 = 5 \) | 2 | Add, then Multiply |
Quick Tips for the ISEE
1
Read carefully: Identify what the question asks for. Sometimes you need to find \( 2x \), not \( x \).
2
Back-solve: On multiple choice, plug each answer into the equation and check which one works.
3
Watch for negatives: Negative coefficients and subtraction are the #1 source of errors.
4
Verify: Substitute your answer back into the original equation before moving on.
Question 1 of 5
Solve for \( x \):
\( x + 13 = 27 \)
Subtract 13 from both sides: \( x = 27 - 13 = 14 \). The inverse of adding 13 is subtracting 13.
Question 2 of 5
Solve for \( n \):
\( 7n = 56 \)
Divide both sides by 7: \( n = \frac{56}{7} = 8 \). The inverse of multiplying by 7 is dividing by 7.
Question 3 of 5
Solve for \( x \):
\( 2x + 9 = 25 \)
First subtract 9: \( 2x = 16 \). Then divide by 2: \( x = 8 \). Check: \( 2(8) + 9 = 25 \) ✓
Question 4 of 5
Solve for \( y \):
\( \dfrac{y}{3} - 4 = 6 \)
Add 4 to both sides: \( \frac{y}{3} = 10 \). Multiply both sides by 3: \( y = 30 \). Check: \( \frac{30}{3} - 4 = 10 - 4 = 6 \) ✓
Question 5 of 5
Solve for \( x \):
\( 15 = -3x + 6 \)
Subtract 6 from both sides: \( 9 = -3x \). Divide by \( -3 \): \( x = -3 \). Watch the negative sign! Check: \( -3(-3) + 6 = 9 + 6 = 15 \) ✓