One-Step
Equations
Learn to isolate a variable using a single inverse operation: the foundation of all equation solving.
What Is an Equation?
An equation is a mathematical statement that says two expressions are equal. The equals sign \( = \) acts like a balance point: whatever is on the left side must have the same value as what is on the right side.
What Is a Variable?
A variable is a letter (like \( x \), \( y \), or \( n \)) that stands in for a number we don't know yet. In a one-step equation, our goal is to figure out what number the variable represents.
What Makes It "One-Step"?
A one-step equation requires exactly one operation to isolate the variable. The variable has only one thing "attached" to it: addition, subtraction, multiplication, or division.
The Golden Rule of Equations
Inverse Operations
To isolate the variable, we use the inverse (opposite) operation to undo whatever is being done to it. Each operation has a partner that cancels it out.
| Operation | Inverse | Example |
|---|---|---|
| Addition (\( + \)) | Subtraction (\( - \)) | \( x + 4 = 9 \) → subtract 4 |
| Subtraction (\( - \)) | Addition (\( + \)) | \( x - 3 = 7 \) → add 3 |
| Multiplication (\( \times \)) | Division (\( \div \)) | \( 5x = 20 \) → divide by 5 |
| Division (\( \div \)) | Multiplication (\( \times \)) | \( \frac{x}{6} = 3 \) → multiply by 6 |
Equations vs Expressions
Students often confuse these two. An expression is a math phrase with no equals sign. An equation always has an equals sign connecting two sides.
No equals sign. You simplify it, but you cannot "solve" it for a single answer.
Has an equals sign. You can solve it to find \( x \).
How to Solve in One Step
Example 1: Addition Equation
Example 2: Subtraction Equation
Example 3: Multiplication Equation
Example 4: Division Equation
Visualizing \( x + 5 = 12 \)
Common Mistakes
Quick Reference Summary
| If You See | Do This | Example | Answer |
|---|---|---|---|
| \( x + a = b \) | Subtract \( a \) | \( x + 9 = 14 \) | \( x = 5 \) |
| \( x - a = b \) | Add \( a \) | \( x - 6 = 11 \) | \( x = 17 \) |
| \( ax = b \) | Divide by \( a \) | \( 7x = 42 \) | \( x = 6 \) |
| \( \frac{x}{a} = b \) | Multiply by \( a \) | \( \frac{x}{5} = 8 \) | \( x = 40 \) |
Question 1 of 5
Solve for \( x \):
\( x + 11 = 25 \)
Question 2 of 5
Solve for \( y \):
\( y - 9 = 17 \)
Question 3 of 5
Solve for \( n \):
\( 6n = 54 \)
Question 4 of 5
Solve for \( k \):
\( \dfrac{k}{7} = 8 \)
Question 5 of 5
A student writes the following to solve \( x + 7 = 20 \):
\( x + 7 + 7 = 20 + 7 \)
What did the student do wrong?