01 / 21
\(x\)
\(7\)
 =

Multi-Step
Equations

Learn to isolate the variable by undoing operations step by step: the key skill for SSAT quantitative success.

SSAT Middle Level

What Is an Equation?

An equation is a mathematical sentence with an equals sign. It states that two expressions have the same value. Think of it like a balance scale: whatever you do to one side, you must do to the other.

\(3x + 5\)
\(20\)
 = Both sides must stay balanced

One-Step vs. Multi-Step

One-Step
Only one operation to undo. Example: \( x + 4 = 10 \). Subtract 4 from both sides and you're done.
Multi-Step
Two or more operations to undo. Example: \( 3x + 5 = 20 \). You need to subtract, then divide.

The Building Blocks

Variable
A letter (like \( x \)) that stands for an unknown number you need to find.
Coefficient
The number multiplied by the variable. In \( 3x \), the coefficient is 3.
Constant
A fixed number with no variable. In \( 3x + 5 = 20 \), both 5 and 20 are constants.

The Golden Rule of Equations

Whatever you do to one side, you must do to the other side.

\(3x + 5 = 20\)
↓ subtract 5 from both sides
\(3x = 15\)
↓ divide both sides by 3
\(x = 5\)

Inverse Operations

To undo an operation, use its opposite (inverse). Each operation has a partner that cancels it out.

Addition ↔ Subtraction
If the equation has \( +5 \), undo it by subtracting 5 from both sides.
Multiplication ↔ Division
If the variable is multiplied by 3, undo it by dividing both sides by 3.

Two Types You'll See

\( ax + b = c \) \( a(x + b) = c \)
Example \( 2x + 7 = 15 \) \( 2(x + 7) = 30 \)
Step 1 Subtract 7 Divide by 2
Step 2 Divide by 2 Subtract 7
Answer \( x = 4 \) \( x = 8 \)

The 4-Step Strategy

1
Simplify each side if needed: combine like terms or distribute.
2
Move constants away from the variable using addition or subtraction.
3
Isolate the variable by dividing or multiplying to remove the coefficient.
4
Check your answer by plugging it back into the original equation.

Example 1: Solve \( 4x + 9 = 25 \)

Start
\(4x + 9 = 25\)
 Subtract 9 from both sides Step 1
\(4x = 16\)
 Divide both sides by 4 Step 2
\(x = 4\)
Check: \(4(4)+9=16+9=25\) ✓

Example 2: Solve \( \frac{x}{3} - 7 = 2 \)

Start
\(\frac{x}{3} - 7 = 2\)
 Add 7 to both sides Step 1
\(\frac{x}{3} = 9\)
 Multiply both sides by 3 Step 2
\(x = 27\)
Check: \(\frac{27}{3}-7=9-7=2\) ✓

Example 3: Solve \( 5(x - 2) + 3 = 28 \)

Start
\(5(x - 2) + 3 = 28\)
 Subtract 3 from both sides Step 1
\(5(x - 2) = 25\)
 Divide both sides by 5 Step 2
\(x - 2 = 5\)
 Add 2 to both sides Step 3
\(x = 7\)
Check: \(5(7-2)+3=5(5)+3=28\) ✓
Alternative: distribute 5 first to get \(5x - 10 + 3 = 28\)

Variables on Both Sides

Sometimes the variable appears on both sides of the equation. Move all variable terms to one side first.

Start
\(5x + 3 = 2x + 18\)
Subtract \(2x\) from both sides
 Step 1
\(3x + 3 = 18\)
 Subtract 3 from both sides Step 2
\(3x = 15\)
 Divide both sides by 3 Step 3
\(x = 5\)
Check: \(5(5)+3=28\) and \(2(5)+18=28\) ✓

Simplify Before Solving

If one side has like terms, combine them first. This keeps your work clean and reduces errors.

Start
\(2x + 3x + 4 = 29\)
Combine \(2x + 3x = 5x\)
 Simplified
\(5x + 4 = 29\)
 Subtract 4 from both sides Step 1
\(5x = 25\)
 Divide both sides by 5 Step 2
\(x = 5\)

Handling Negative Numbers

Don't panic when you see negatives. The same rules apply: use inverse operations and keep the equation balanced.

Start
\(-2x + 10 = 4\)
 Subtract 10 from both sides Step 1
\(-2x = -6\)
Divide both sides by \(-2\)
 Step 2
\(x = 3\)
Check: \(-2(3)+10=-6+10=4\) ✓

Common Mistakes to Avoid

✕ Forgetting Both Sides
Subtracting from only one side. If you subtract 5 from the left, you must subtract 5 from the right too.
✕ Wrong Order
Dividing before subtracting. Always undo addition/subtraction first, then multiplication/division.
✕ Sign Errors
Dropping a negative sign. Write every step clearly and watch your signs when distributing or subtracting.

Quick Reference Table

Equation Type Example Key Steps Answer
Two-step \( 3x + 7 = 22 \) Subtract, divide \( x = 5 \)
With fractions \( \frac{x}{4} - 3 = 5 \) Add, multiply \( x = 32 \)
With parentheses \( 2(x + 6) = 20 \) Divide, subtract \( x = 4 \)
Variables both sides \( 4x + 1 = x + 10 \) Move \( x \), subtract, divide \( x = 3 \)
Combine like terms \( x + 2x + 5 = 20 \) Combine, subtract, divide \( x = 5 \)

Question 1 of 5

Solve for \( x \): \( 2x + 8 = 22 \)

Question 2 of 5

Solve for \( x \): \( \frac{x}{5} + 3 = 10 \)

Question 3 of 5

Solve for \( x \): \( 3(x + 4) = 27 \)

Question 4 of 5

Solve for \( x \): \( 6x - 4 = 2x + 12 \)

Question 5 of 5

Solve for \( x \): \( -3x + 15 = 0 \)