One-Variable Equations - Seventh Grade
One-Step, Two-Step, Properties of Equality & Word Problems
1. Understanding Equations
What is an Equation?
An equation is a mathematical statement that
two expressions are EQUAL (uses = sign)
• Example: 2x + 3 = 9
• Contains a variable (unknown value)
• Goal: Find the value of the variable
Solution of an Equation
A solution is the value of the variable
that makes the equation TRUE
Example: In x + 5 = 12, the solution is x = 7
Check: 7 + 5 = 12 ✓
Which x Satisfies an Equation?
Test by substituting: Which value satisfies x + 3 = 10?
Options: x = 5, x = 7, x = 13
Test x = 5: 5 + 3 = 8 ✗
Test x = 7: 7 + 3 = 10 ✓
Test x = 13: 13 + 3 = 16 ✗
Answer: x = 7 satisfies the equation
2. Properties of Equality
Addition Property of Equality
If a = b, then a + c = b + c
Add the SAME value to both sides
The equation stays balanced
Subtraction Property of Equality
If a = b, then a − c = b − c
Subtract the SAME value from both sides
Multiplication Property of Equality
If a = b, then a × c = b × c
Multiply both sides by the SAME value
Division Property of Equality
If a = b, then a ÷ c = b ÷ c (c ≠ 0)
Divide both sides by the SAME value (not zero)
Key Principle
Whatever you do to one side,
do to the OTHER side!
3. Solving One-Step Equations
What is a One-Step Equation?
An equation that can be solved in
ONE step using inverse operations
Examples: x + 5 = 12, 3x = 15, x/4 = 2
Inverse Operations
| Operation | Inverse Operation |
|---|---|
| Addition (+) | Subtraction (−) |
| Subtraction (−) | Addition (+) |
| Multiplication (×) | Division (÷) |
| Division (÷) | Multiplication (×) |
Type 1: Addition Equations (x + a = b)
Solve: x + 7 = 15
Step 1: Subtract 7 from both sides
x + 7 − 7 = 15 − 7
x = 8
Check: 8 + 7 = 15 ✓
Solution: x = 8
Type 2: Subtraction Equations (x − a = b)
Solve: x − 4 = 9
Step 1: Add 4 to both sides
x − 4 + 4 = 9 + 4
x = 13
Check: 13 − 4 = 9 ✓
Solution: x = 13
Type 3: Multiplication Equations (ax = b)
Solve: 5x = 20
Step 1: Divide both sides by 5
5x ÷ 5 = 20 ÷ 5
x = 4
Check: 5(4) = 20 ✓
Solution: x = 4
Type 4: Division Equations (x/a = b)
Solve: x/3 = 6
Step 1: Multiply both sides by 3
x/3 × 3 = 6 × 3
x = 18
Check: 18/3 = 6 ✓
Solution: x = 18
With Decimals and Fractions
Example 1: x + 2.5 = 7.8
x = 7.8 − 2.5 = 5.3
Example 2: x/2 = 3/4
x = 3/4 × 2 = 3/2 or 1.5
4. Solving Two-Step Equations
What is a Two-Step Equation?
An equation that requires TWO steps to solve
Form: ax + b = c
Example: 3x + 5 = 14
Steps to Solve Two-Step Equations
Step 1: Add or subtract to isolate the term with the variable
Step 2: Multiply or divide to solve for the variable
Step 3: Check your answer!
Example 1: Without Parentheses
Solve: 2x + 3 = 11
Step 1: Subtract 3 from both sides
2x + 3 − 3 = 11 − 3
2x = 8
Step 2: Divide both sides by 2
2x ÷ 2 = 8 ÷ 2
x = 4
Check: 2(4) + 3 = 8 + 3 = 11 ✓
Solution: x = 4
Example 2: With Parentheses
Solve: 3(x + 2) = 15
Step 1: Use distributive property
3x + 6 = 15
Step 2: Subtract 6 from both sides
3x = 9
Step 3: Divide both sides by 3
x = 3
Check: 3(3 + 2) = 3(5) = 15 ✓
Solution: x = 3
Example 3: With Fractions
Solve: x/2 + 3 = 7
Step 1: Subtract 3 from both sides
x/2 = 4
Step 2: Multiply both sides by 2
x = 8
Solution: x = 8
Example 4: Negative Coefficient
Solve: −2x + 5 = 13
Subtract 5: −2x = 8
Divide by −2: x = −4
Solution: x = −4
5. Solving Equations with Like Terms
Strategy
Step 1: Combine like terms on each side FIRST
Step 2: Then solve as a regular equation
Example
Solve: 3x + 2x − 4 = 11
Step 1: Combine like terms (3x + 2x)
5x − 4 = 11
Step 2: Add 4 to both sides
5x = 15
Step 3: Divide by 5
x = 3
Solution: x = 3
6. Writing Equations from Words
Key Translation Words
"is" or "equals" → =
"more than" or "sum" → +
"less than" or "difference" → −
"times" or "product" → ×
"divided by" or "quotient" → ÷
Examples
Example 1: "Five more than a number is 12"
A number → x
Five more than → + 5
is → =
Equation: x + 5 = 12
Example 2: "Three times a number minus 7 equals 20"
Three times a number → 3x
minus 7 → − 7
equals → =
Equation: 3x − 7 = 20
7. Two-Step Equations: Word Problems
Steps to Solve
Step 1: Read and understand the problem
Step 2: Define the variable (let x = ...)
Step 3: Write the equation
Step 4: Solve the equation
Step 5: Answer the question
Example
Problem: Maria has some money. She spent $15 and has $42 left. How much money did she have originally?
Step 1: Let x = original amount
Step 2: Write equation
x − 15 = 42
Step 3: Solve
x = 42 + 15
x = 57
Answer: Maria had $57 originally
Another Example
Problem: A gym charges a $25 membership fee plus $5 per visit. If Tom paid $60, how many times did he visit?
Let x = number of visits
Equation: 5x + 25 = 60
5x = 35
x = 7
Answer: Tom visited 7 times
Quick Reference: Solving Equations
| Equation Type | Example | Inverse Operation |
|---|---|---|
| x + a = b | x + 5 = 12 | Subtract a |
| x − a = b | x − 3 = 7 | Add a |
| ax = b | 4x = 20 | Divide by a |
| x/a = b | x/2 = 5 | Multiply by a |
| ax + b = c | 3x + 4 = 13 | Subtract b, then divide by a |
💡 Important Tips to Remember
✓ Golden Rule: Whatever you do to one side, do to the other!
✓ Inverse operations: Undo the operation acting on the variable
✓ One-step equations: Use ONE inverse operation
✓ Two-step equations: Add/subtract FIRST, then multiply/divide
✓ With parentheses: Use distributive property first
✓ With like terms: Combine them FIRST
✓ Always check: Substitute your answer back into the original equation
✓ Negative coefficients: Remember to divide by negative number
✓ Word problems: Define your variable clearly
✓ Work backwards: Think about what operations were done to x
🧠 Memory Tricks & Strategies
Golden Rule:
"What you do to one side, do to the other - keep the equation balanced like brother and brother!"
Inverse Operations:
"Addition's buddy is subtraction, Multiplication's friend is division - they're opposite in action!"
Two-Step Order:
"Add or subtract to set it free, then multiply or divide to find x's key!"
Checking Your Answer:
"Plug it back, see if it's true - if both sides match, your work is through!"
Word Problems:
"Read it twice, define x once - write the equation and solve with confidence!"
Order Matters:
"In two-step solving, here's the trick - undo addition/subtraction first, then multiply/divide quick!"
Master One-Variable Equations! 🔢 ⚖️
Remember: Keep the equation balanced and use inverse operations!