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\(x\)
\(=\)

Two-Step
Equations

Learn to isolate the variable by undoing two operations, then verify your answer with substitution.

Pre-Algebra

What Is an Equation?

An equation is a mathematical sentence that says two expressions are equal. Think of a balance scale: whatever you do to one side, you must do to the other to keep it balanced.

\(2x + 3\)
\(=\)
\(11\)

One-Step vs Two-Step

One-Step
Only one operation to undo.
Example: \( x + 5 = 12 \)
Two-Step
Two operations to undo.
Example: \( 2x + 5 = 12 \)

Why Do Two-Step Equations Matter?

Real-world problems almost never involve just one operation. Suppose you earn \( \$8 \) per hour and already saved \( \$20 \). How many hours do you need to reach \( \$100 \)?

Mastering two-step equations is the foundation for multi-step equations, inequalities, and systems of equations later on.

The General Form

A two-step equation looks like:

\( a \)
The coefficient: multiplies the variable
\( b \)
The constant: added to or subtracted from the variable term
\( c \)
The result: the value the expression equals

The Golden Rule of Equations

Left Side Right Side
\(=\)
same operation same operation

This keeps the equation balanced. We use inverse operations to "peel away" layers around the variable.

Inverse Operations

Each operation has an opposite that undoes it:

Operation Inverse
Addition \( (+) \) Subtraction \( (-) \)
Subtraction \( (-) \) Addition \( (+) \)
Multiplication \( (\times) \) Division \( (\div) \)
Division \( (\div) \) Multiplication \( (\times) \)

Order Matters: Reverse PEMDAS

When the equation was built, multiplication happened first, then addition. To undo it, reverse the order.

Building \( 2x + 3 = 11 \)
Start with \( x \) → multiply by 2 → add 3 → get 11
Solving (reverse)
Start with 11 → subtract 3 → divide by 2 → get \( x \)

How to Solve in 4 Steps

1
Identify the two operations acting on the variable.
2
Undo addition/subtraction on both sides (isolate the variable term).
3
Undo multiplication/division on both sides (isolate the variable).
4
Check by substituting your answer back into the original equation.

Example 1: Solve \( 3x + 5 = 20 \)

\(3x + 5 = 20\)
 - 5 both sides
\(3x = 15\)
 ÷ 3 both sides
\(x = 5\)
Check: \(3(5) + 5 = 15 + 5 = 20\) ✓ Matches the right side!

Example 2: Solve \( 4x - 7 = 25 \)

\(4x - 7 = 25\)
 + 7 both sides
\(4x = 32\)
 ÷ 4 both sides
\(x = 8\)
Check: \(4(8) - 7 = 32 - 7 = 25\) ✓ It balances!

Example 3: Solve \( \frac{x}{3} + 4 = 10 \)

\(\frac{x}{3} + 4 = 10\)
 - 4 both sides
\(\frac{x}{3} = 6\)
 × 3 both sides
\(x = 18\)
Check: \(\frac{18}{3} + 4 = 6 + 4 = 10\) ✓ Perfect!

Example 4: Negative Solutions

Solve \( 5x + 12 = 2 \)

1
Subtract 12 from both sides: \( 5x = 2 - 12 = -10 \)
2
Divide by 5 on both sides: \( x = \frac{-10}{5} = -2 \)
3
Check: \( 5(-2) + 12 = -10 + 12 = 2 \) ✓

Always Check Your Work

After solving, substitute your answer back into the original equation. If both sides are equal, your solution is correct.

ORIGINAL \(3x + 5 = 20\)
SUBSTITUTE x = 5 \(3(5) + 5 = ?\)
SIMPLIFY \(20 = 20\) ✓

Common Mistakes

✕ Wrong Order
Dividing before subtracting. Always undo addition/subtraction first.
✕ One Side Only
Subtracting from one side but forgetting the other. Both sides must change.
✕ Sign Errors
Losing a negative sign, especially when subtracting a larger number. Double-check signs.

Summary Table

Equation Type Step 1 Step 2 Example
\( ax + b = c \) Subtract \( b \) Divide by \( a \) \( 2x + 3 = 11 \)
\( ax - b = c \) Add \( b \) Divide by \( a \) \( 4x - 7 = 25 \)
\( \frac{x}{a} + b = c \) Subtract \( b \) Multiply by \( a \) \( \frac{x}{3} + 4 = 10 \)
\( \frac{x}{a} - b = c \) Add \( b \) Multiply by \( a \) \( \frac{x}{5} - 2 = 6 \)

Question 1 of 5

Solve: \( 2x + 7 = 19 \)

Question 2 of 5

Solve: \( 5x - 3 = 22 \)

Question 3 of 5

Solve: \( \frac{x}{4} + 6 = 11 \)

Question 4 of 5

A student solves \( 3x + 9 = 24 \) and gets \( x = 11 \). What did they do wrong?

Question 5 of 5

Solve: \( 6x + 14 = 2 \)