One-Variable Equations - Sixth Grade
Complete Notes & Formulas
1. Expressions vs Equations
Key Difference
| Type | Has = Sign? | Example | Can Solve? |
|---|---|---|---|
| Expression | NO ✗ | 3x + 5 | Can simplify only |
| Equation | YES ✓ | 3x + 5 = 14 | Can solve for x |
An EQUATION is a mathematical statement
that says two expressions are EQUAL
Left Side = Right Side
2. Does x Satisfy an Equation?
What Does "Satisfy" Mean?
A value SATISFIES an equation if:
Substituting it makes BOTH SIDES EQUAL
Steps to Check
Step 1: Substitute the value for the variable
Step 2: Simplify the left side
Step 3: Simplify the right side
Step 4: Check if both sides are equal
Example: Does x = 5 satisfy x + 7 = 12?
Step 1: Substitute x = 5
(5) + 7 = 12
Step 2: Simplify left side
12 = 12
Step 3: Compare
Both sides are equal!
Answer: YES, x = 5 satisfies the equation ✓
Example: Does x = 8 satisfy 2x = 20?
Substitute x = 8: 2(8) = 20
Simplify: 16 = 20
16 ≠ 20 (Not equal!)
Answer: NO, x = 8 does NOT satisfy ✗
3. Inverse Operations
What are Inverse Operations?
Inverse operations are OPPOSITE operations
They UNDO each other!
Inverse Operation Pairs
| Operation | Inverse Operation | Example |
|---|---|---|
| Addition (+) | Subtraction (−) | x + 5 → subtract 5 |
| Subtraction (−) | Addition (+) | x − 3 → add 3 |
| Multiplication (×) | Division (÷) | 5x → divide by 5 |
| Division (÷) | Multiplication (×) | x/4 → multiply by 4 |
4. Solving One-Step Addition Equations
Rule
If a number is ADDED to the variable
SUBTRACT that number from BOTH SIDES
x + a = b → x = b − a
Example 1: Solve x + 7 = 15
x + 7 = 15
x + 7 − 7 = 15 − 7 (Subtract 7 from both sides)
x = 8
Answer: x = 8
Check: 8 + 7 = 15 ✓
Example 2: Solve y + 3.5 = 10.2
y + 3.5 = 10.2
y + 3.5 − 3.5 = 10.2 − 3.5
y = 6.7
Answer: y = 6.7
5. Solving One-Step Subtraction Equations
Rule
If a number is SUBTRACTED from the variable
ADD that number to BOTH SIDES
x − a = b → x = b + a
Example 1: Solve x − 9 = 12
x − 9 = 12
x − 9 + 9 = 12 + 9 (Add 9 to both sides)
x = 21
Answer: x = 21
Check: 21 − 9 = 12 ✓
Example 2: Solve n − 1/4 = 3/4
n − 1/4 = 3/4
n − 1/4 + 1/4 = 3/4 + 1/4
n = 4/4 = 1
Answer: n = 1
6. Solving One-Step Multiplication Equations
Rule
If the variable is MULTIPLIED by a number
DIVIDE both sides by that number
ax = b → x = b ÷ a
Example 1: Solve 6x = 42
6x = 42
6x ÷ 6 = 42 ÷ 6 (Divide both sides by 6)
x = 7
Answer: x = 7
Check: 6(7) = 42 ✓
Example 2: Solve 0.5y = 4
0.5y = 4
0.5y ÷ 0.5 = 4 ÷ 0.5
y = 8
Answer: y = 8
7. Solving One-Step Division Equations
Rule
If the variable is DIVIDED by a number
MULTIPLY both sides by that number
x/a = b → x = b × a
Example 1: Solve x/5 = 8
x/5 = 8
x/5 × 5 = 8 × 5 (Multiply both sides by 5)
x = 40
Answer: x = 40
Check: 40/5 = 8 ✓
Example 2: Solve y/3 = 2.5
y/3 = 2.5
y/3 × 3 = 2.5 × 3
y = 7.5
Answer: y = 7.5
8. Writing Equations from Words
Key Words for Operations
| Operation | Key Words |
|---|---|
| Addition (+) | sum, plus, more than, increased by, total, added to |
| Subtraction (−) | difference, minus, less than, decreased by, fewer, subtracted from |
| Multiplication (×) | product, times, multiplied by, of |
| Division (÷) | quotient, divided by, per, ratio, each |
| Equals (=) | is, equals, is equal to, results in, gives, yields |
Example 1: "A number increased by 12 equals 25"
• "A number" → x
• "increased by 12" → + 12
• "equals 25" → = 25
Equation: x + 12 = 25
Example 2: "Four times a number is 32"
• "Four times a number" → 4x
• "is 32" → = 32
Equation: 4x = 32
9. Solving One-Step Equation Word Problems
Steps for Word Problems
Step 1: Read the problem carefully
Step 2: Identify the unknown (variable)
Step 3: Write the equation
Step 4: Solve using inverse operations
Step 5: Check your answer
Example 1: Addition Problem
Problem: Maria has some marbles. Her friend gives her 15 more marbles. Now she has 42 marbles. How many did she start with?
Let: x = marbles she started with
Equation: x + 15 = 42
Solve: x + 15 − 15 = 42 − 15
x = 27
Answer: Maria started with 27 marbles
Example 2: Multiplication Problem
Problem: Each box contains 8 pencils. If there are 56 pencils total, how many boxes are there?
Let: x = number of boxes
Equation: 8x = 56
Solve: 8x ÷ 8 = 56 ÷ 8
x = 7
Answer: There are 7 boxes
10. Solving Equations with Integers
Same Rules Apply!
Use inverse operations
Be careful with negative signs
Remember: subtracting a negative = adding
Example 1: x + (−5) = 12
x + (−5) = 12
x − 5 = 12 (Add 5 to both sides)
x − 5 + 5 = 12 + 5
x = 17
Answer: x = 17
Example 2: −3x = 15
−3x = 15
−3x ÷ (−3) = 15 ÷ (−3) (Divide both sides by −3)
x = −5
Answer: x = −5
Quick Reference: One-Step Equations
| Equation Type | Example | Operation to Use | Solution Form |
|---|---|---|---|
| Addition | x + 5 = 12 | Subtract 5 | x = 7 |
| Subtraction | x − 3 = 10 | Add 3 | x = 13 |
| Multiplication | 4x = 20 | Divide by 4 | x = 5 |
| Division | x/6 = 7 | Multiply by 6 | x = 42 |
💡 Important Tips to Remember
✓ Equation has = sign, expression does NOT
✓ To check solution: substitute and verify both sides equal
✓ Use inverse operations to isolate the variable
✓ Do same operation to BOTH sides
✓ Addition ↔ Subtraction are inverses
✓ Multiplication ↔ Division are inverses
✓ Goal: Get variable alone on one side
✓ Always check your answer by substituting back
✓ Key words help translate words to equations
✓ Be careful with negatives when solving with integers
🧠 Memory Tricks & Strategies
Equation vs Expression:
"Equation has equal sign - that's how you know, it's equation time!"
Inverse Operations:
"Add and subtract, multiply and divide - inverse pairs help solutions arrive!"
Solving Steps:
"What's done to x, undo it fast - use the inverse, get x at last!"
Checking Solutions:
"Substitute back, see if it's true - if both sides match, you got it through!"
Word Problems:
"Read it twice, find the unknown - write equation, solution shown!"
Master One-Variable Equations! 🔢 = ✓
Remember: Use inverse operations to solve!