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\( x \)
\( = \)

One-Step
Equations

Learn to solve equations with a single inverse operation: addition, subtraction, multiplication, or division.

Pre-Algebra

What Is an Equation?

An equation is a mathematical sentence that says 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.

\( x + 3 \)
\( 10 \)
\( = \)

Variables and Constants

Variable
A letter (like \( x \), \( n \), or \( y \)) that represents an unknown value you need to find.
Constant
A fixed number whose value does not change, like \( 5 \), \( -3 \), or \( \frac{1}{2} \).

What Makes It "One-Step"?

A one-step equation needs exactly one inverse operation to isolate the variable. There are four types, one for each basic operation:

\( x + a = b \)
Subtract \( a \)
\( x - a = b \)
Add \( a \)
\( ax = b \)
Divide by \( a \)
\( \frac{x}{a} = b \)
Multiply by \( a \)

The Inverse Operation Rule

Every operation has an inverse (opposite) that "undoes" it. To solve, apply the inverse of whatever operation is attached to the variable.

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

The Golden Rule of Equations

Original:
\( x + 5 \)
\( = \)
\( 12 \)
-5 both sides
Result:
\( x \)
\( = \)
\( 7 \)

Expressions vs Equations

Expression Equation
Definition Numbers, variables, and operations combined Two expressions connected by \( = \)
Example \( 3x + 7 \) \( 3x + 7 = 22 \)
Equals sign? No Yes
Can you solve? Simplify only Find the unknown

How to Solve in One Step

1
Identify the operation attached to the variable (\( + \), \( - \), \( \times \), or \( \div \)).
2
Choose the inverse operation that will undo it.
3
Apply it to both sides of the equation to keep balance.
4
Simplify to get the variable alone on one side.
5
Check by substituting back into the original equation.

Example 1: Addition Equation

Solve: \( x + 9 = 14 \)

Step 1: Identify
\(9\) is being added to \(x\) → inverse is subtract \(9\)
Step 2: Apply to both sides
\( x + 9 - 9 = 14 - 9 \)
Step 3: Simplify
\( x = 5 \)

Example 2: Subtraction Equation

Solve: \( n - 7 = 15 \)

Step 1: Identify
\(n\) has \(7\) subtracted → inverse is add \(7\)
Step 2: Apply to both sides
\( n - 7 + 7 = 15 + 7 \)
Step 3: Simplify
\( n = 22 \)

Example 3: Multiplication Equation

Solve: \( 4y = 32 \)

Step 1: Identify
\(y\) is multiplied by \(4\) → inverse is divide by \(4\)
Step 2: Apply to both sides
\( \frac{4y}{4} = \frac{32}{4} \)
Step 3: Simplify
\( y = 8 \)

Example 4: Division Equation

Solve: \( \dfrac{m}{6} = 9 \)

Step 1: Identify
\(m\) is divided by \(6\) → inverse is multiply by \(6\)
Step 2: Apply to both sides
\( \frac{m}{6} \times 6 = 9 \times 6 \)
Step 3: Simplify
\( m = 54 \)

Trickier: Negative Numbers

Solve: \( x + 12 = -3 \)

1
Operation: \( x \) has \( 12 \) added to it.
2
Inverse: Subtract \( 12 \) from both sides.
3
Apply: \( x + 12 - 12 = -3 - 12 \)
4
Simplify: \( x = -15 \)

Trickier: Fractions

Solve: \( \dfrac{x}{5} = \dfrac{3}{10} \)

1
Operation: \( x \) is divided by \( 5 \).
2
Inverse: Multiply both sides by \( 5 \).
3
Apply: \( \dfrac{x}{5} \times 5 = \dfrac{3}{10} \times 5 \)
4
Simplify: \( x = \dfrac{15}{10} = \dfrac{3}{2} \)

Common Mistakes

✕ Wrong Inverse
Adding when you should subtract, or multiplying when you should divide. Always ask: "What undoes this operation?"
✕ One Side Only
Performing the inverse on just one side of the equation. Both sides must stay balanced.
✕ Sign Errors
Forgetting the negative sign when subtracting larger numbers. For example, \( -3 - 12 = -15 \), not \( 9 \).

Summary of One-Step Equations

Equation Form Inverse Operation Example
\( x + a = b \) Subtract \( a \) \( x + 9 = 14 \implies x = 5 \)
\( x - a = b \) Add \( a \) \( n - 7 = 15 \implies n = 22 \)
\( ax = b \) Divide by \( a \) \( 4y = 32 \implies y = 8 \)
\( \frac{x}{a} = b \) Multiply by \( a \) \( \frac{m}{6} = 9 \implies m = 54 \)

Question 1 of 5

Solve: \( x + 7 = 20 \)

Question 2 of 5

Solve: \( n - 11 = 4 \)

Question 3 of 5

Solve: \( 6p = 42 \)

Question 4 of 5

Solve: \( \dfrac{w}{8} = 5 \)

Question 5 of 5

Solve: \( t + 15 = -2 \)