Generate
Equivalent Expressions
Learn to use the distributive property, combining like terms, and properties of operations to rewrite expressions in equivalent forms.
6th Grade Math
What Are Equivalent Expressions?
Two expressions are equivalent if they have the same value for every possible value of the variable. They may look different, but they always produce the same result.
Why Rewrite Expressions?
Simplify
Make complex expressions shorter and easier to work with.
Solve
Rewriting helps isolate variables when solving equations.
Reveal Structure
See patterns like common factors that aren't obvious at first.
Three Key Properties
1
Distributive Property: \( a(b + c) = ab + ac \). Multiply each term inside the parentheses by the factor outside.
2
Commutative Property: \( a + b = b + a \) and \( a \cdot b = b \cdot a \). Order doesn't change the result.
3
Combining Like Terms: \( y + y + y = 3y \). Add coefficients of terms that share the same variable part.
The Distributive Property: Expanding
The Distributive Property: Factoring
Factoring is the reverse of distributing. Instead of expanding parentheses, you pull out a common factor to create them.
Combining Like Terms
Like terms have the same variable raised to the same power. You can add or subtract their coefficients to simplify.
Expanding vs. Factoring
| Expanding | Factoring | |
|---|---|---|
| Direction | Remove parentheses | Create parentheses |
| Operation | Multiply each term | Divide out the GCF |
| Example Start | \( 5(x + 3) \) | \( 10x + 15 \) |
| Example Result | \( 5x + 15 \) | \( 5(2x + 3) \) |
| When to Use | Simplify for solving | Reveal common factors |
How to Generate Equivalent Expressions
1
Identify the operation: Is there a factor outside parentheses (expand) or a common factor across terms (factor)?
2
Apply the property: Use the distributive property to expand or factor. Use commutative/associative properties to rearrange.
3
Combine like terms: Group terms with the same variable and add their coefficients.
4
Verify: Plug in a test value (like \( x = 2 \)) into both expressions to confirm they give the same result.
Example 1: Expanding
Expand: \( 4(3x + 5) \)
Example 2: Factoring
Factor: \( 15x + 10 \)
1
Find the GCF of 15 and 10. Factors of 15: 1, 3, 5, 15. Factors of 10: 1, 2, 5, 10. GCF = 5.
2
Divide each term by 5: \( 15x \div 5 = 3x \) and \( 10 \div 5 = 2 \).
3
Write the factored form: \( 5(3x + 2) \).
Example 3: Combining Like Terms
Simplify: \( 3x + 7 + 2x + 4 \)
Example 4: Multi-Step Simplification
Simplify: \( 2(x + 4) + 3x \)
1
Distribute the 2: \( 2 \cdot x + 2 \cdot 4 + 3x = 2x + 8 + 3x \)
2
Combine like terms: \( 2x + 3x = 5x \), so we get \( 5x + 8 \).
3
Verify: Let \( x = 3 \). Original: \( 2(3+4) + 3(3) = 14 + 9 = 23 \). Simplified: \( 5(3) + 8 = 23 \). ✓
Common Mistakes
✕ Incomplete Distribution
Writing \( 3(x + 4) = 3x + 4 \) instead of \( 3x + 12 \). You must multiply every term inside.
✕ Combining Unlike Terms
Adding \( 5x + 3 = 8x \). The constant \(3\) and the variable term \( 5x \) are not like terms.
✕ Wrong GCF
Factoring \( 12x + 8 \) as \( 2(6x + 4) \) instead of \( 4(3x + 2) \). Always use the greatest common factor.
Summary of Techniques
| Technique | What You Do | Example |
|---|---|---|
| Expand | Multiply out parentheses | \( 3(x+2) = 3x+6 \) |
| Factor | Pull out the GCF | \( 12x+8 = 4(3x+2) \) |
| Combine Like Terms | Add matching coefficients | \( 4x+2x = 6x \) |
| Commutative | Reorder terms | \( 5+x = x+5 \) |
Question 1 of 5
Which expression is equivalent to \( 5(2x + 3) \)?
Apply the distributive property: \( 5 \cdot 2x + 5 \cdot 3 = 10x + 15 \). You must multiply both terms inside the parentheses by \(5\).
Question 2 of 5
What is the factored form of \( 18x + 12 \)?
The GCF of \(18\) and \(12\) is \(6\). Dividing: \( 18x \div 6 = 3x \) and \( 12 \div 6 = 2 \). So: \( 6(3x + 2) \). The other options use smaller factors, not the greatest common factor.
Question 3 of 5
Simplify: \( 4x + 3 + 2x + 7 \)
Combine the x-terms: \( 4x + 2x = 6x \). Combine the constants: \( 3 + 7 = 10 \). Result: \( 6x + 10 \).
Question 4 of 5
Which expression is equivalent to \( x + x + x + x \)?
Adding \( x \) four times means \( 4 \cdot x = 4x \). Be careful: \( x^4 \) means \( x \cdot x \cdot x \cdot x \) (multiplying, not adding), which is very different.
Question 5 of 5
Simplify: \( 3(2x + 1) + 4x \)
First distribute: \( 3(2x+1) = 6x + 3 \). Then combine with \( 4x \): \( 6x + 3 + 4x = 10x + 3 \).