01 / 20
\(x\)
\(7\)

One- and Two-Step
Equations

Master the art of isolating variables using inverse operations: the foundation of all algebra.

ISEE Middle Level Quantitative Reasoning

What Is an Equation?

An equation is a mathematical sentence with an equals sign that says two expressions have the same value. Think of it as a perfectly balanced scale.

\(x + 3\)
\(10\)
\(=\)
 Both sides must stay balanced

Variables and Constants

Variable
A letter (like \( x \), \( y \), or \( n \)) that represents an unknown value we want to find.
Constant
A fixed number (like \( 5 \), \( -3 \), or \( 12 \)) whose value never changes.
Coefficient
The number multiplied by the variable. In \( 4x \), the coefficient is 4.

Why Do Equations Matter?

Equations let us translate real-world problems into math. Once you write an equation, solving it gives you the answer.

1
Translate: Turn a word problem into an equation with a variable.
2
Solve: Use inverse operations to isolate the variable.
3
Check: Plug your answer back in to verify both sides are equal.

The Inverse Operations Rule

Every operation has an opposite that "undoes" it. To isolate the variable, apply the inverse operation to both sides.

Addition \(+\)
Subtraction \(-\)
 undo undo
Multiplication \(\times\)
Division \(\div\)
 undo undo

One-Step vs Two-Step

One-Step Two-Step
Operations to undo 1 2
Example \( x + 5 = 12 \) \( 3x + 5 = 20 \)
Strategy One inverse operation Undo addition/subtraction first, then multiplication/division
Moves 1 move 2 moves

How to Solve Step by Step

1
Identify the variable and what operations are applied to it.
2
Undo addition or subtraction first: apply the inverse to both sides.
3
Undo multiplication or division next: apply the inverse to both sides.
4
Check your answer by substituting back into the original equation.

Example 1: One-Step (Addition)

Solve: \( x + 9 = 17 \)

Step 1: Identify the operation
\(x + 9 = 17\)
 (+9) Step 2: Subtract 9 from both sides
\(x + 9 - 9 = 17 - 9\)
 (-9) Step 3: Simplify
\(x = 8\)

Example 2: One-Step (Multiplication)

Solve: \( 5x = 35 \)

Step 1: Identify the operation
\(5x = 35\)
 (×5) Step 2: Divide both sides by 5
\(\frac{5x}{5} = \frac{35}{5}\)
 (÷5) Step 3: Simplify
\(x = 7\)

Example 3: Two-Step Equation

Solve: \( 3x + 7 = 22 \)

Original equation
\(3x + 7 = 22\)
 Step 1: Subtract 7 from both sides
\(3x + 7 - 7 = 22 - 7\)
 (-7)
\(3x = 15\)
 Step 2: Divide both sides by 3
\(x = 5\)
 (÷3)

Example 4: Two-Step with Subtraction

Solve: \( \dfrac{x}{4} - 3 = 5 \)

1
Add 3 to both sides: \( \dfrac{x}{4} - 3 + 3 = 5 + 3 \) which gives \( \dfrac{x}{4} = 8 \)
2
Multiply both sides by 4: \( \dfrac{x}{4} \times 4 = 8 \times 4 \) which gives \( x = 32 \)

Example 5: Negative Coefficients

Solve: \( -2x + 11 = 3 \)

Original equation
\(-2x + 11 = 3\)
 Step 1: Subtract 11 from both sides
\(-2x + 11 - 11 = 3 - 11\)
\(-2x = -8\)
 Step 2: Divide both sides by -2
\(x = 4\)

Common Mistakes to Avoid

✕ Wrong Order
Dividing before subtracting in a two-step equation. Always undo addition/subtraction first.
✕ One-Sided Operation
Applying an operation to only one side of the equation. Both sides must stay balanced.
✕ Sign Errors
Forgetting that subtracting a negative becomes addition, or that \( (-) \div (-) = (+) \).

Equations with Fractions

When the variable is divided by a number, multiply both sides by that number to undo the fraction.

1
\( \dfrac{x}{6} = 9 \): Multiply both sides by 6 to get \( x = 54 \)
2
\( \dfrac{x}{3} + 4 = 10 \): Subtract 4 first: \( \dfrac{x}{3} = 6 \). Then multiply by 3: \( x = 18 \)

Quick Reference Guide

Equation Type Example Steps Solution
Add/Subtract \( x + 8 = 15 \) Subtract 8 \( x = 7 \)
Multiply \( 6x = 42 \) Divide by 6 \( x = 7 \)
Divide \( \frac{x}{5} = 4 \) Multiply by 5 \( x = 20 \)
Two-Step \( 2x + 3 = 11 \) Subtract 3, divide by 2 \( x = 4 \)
Negative Coeff. \( -4x + 1 = 9 \) Subtract 1, divide by \(-4\) \( x = -2 \)

Question 1 of 5

Solve: \( x - 14 = 23 \)

Question 2 of 5

Solve: \( 7x = 56 \)

Question 3 of 5

Solve: \( 4x + 9 = 33 \)

Question 4 of 5

Solve: \( \dfrac{x}{5} - 2 = 6 \)

Question 5 of 5

Solve: \( -3x + 10 = -5 \)