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3. Equivalent Expressions

# Equivalent Expressions

Equivalent expressions are algebraic expressions that may look different but represent the same quantity or value. Understanding equivalent expressions is fundamental in algebra and is a key skill tested in the Digital SAT exam. This topic involves simplifying expressions, factoring, and using properties of operations to rewrite expressions. Mastery of this concept helps in solving equations and understanding algebraic relationships.

## Learning Objectives

By studying equivalent expressions, you will be able to identify and create equivalent expressions in one variable, apply properties of operations (such as distributive, associative, and commutative properties), and simplify algebraic expressions. They will also learn to use equivalent expressions to solve equations and real-world problems, enhancing their algebraic thinking and problem-solving skills.

## Understanding Equivalent Expressions

Equivalent expressions are two or more expressions that have the same value for all values of the variables involved. For instance, 3(x+2) and 3x+6 are equivalent because they yield the same result for any value of x.

#### Properties of Operations

To create and identify equivalent expressions, it’s essential to use the properties of operations. Here are some key properties:

Distributive Property

The distributive property allows you to multiply a single term by each term inside a parenthesis:
a(b+c)=ab+ac
Example: 2(x+3)=2x+6

Associative Property

The associative property states that the way terms are grouped does not change their sum or product:
(a+b)+c=a+(b+c)
(ab)c=a(bc)

Commutative Property

The commutative property indicates that the order of terms does not affect their sum or product:
a+b=b+a
ab=ba

### Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and using the properties of operations to rewrite the expression in a simpler form.

Combining Like Terms

Like terms are terms that have the same variables raised to the same power. Combine them by adding or subtracting their coefficients.
Example:
3x+5x=(3+5)x=8x
4y−2y=(4−2)y=2y

### Factoring Expressions

Factoring is the process of breaking down an expression into a product of simpler expressions. This is often the reverse of using the distributive property.

Common Factoring Techniques

• Factoring out the Greatest Common Factor (GCF): Identify the GCF of all terms and factor it out.
Example: 6x+9=3(2x+3)
• Factoring Quadratic Expressions: For expressions of the form ax2+bx+c, find two numbers that multiply to ac and add to b.
Example: x2+5x+6=(x+2)(x+3)

### Using Equivalent Expressions to Solve Equations

Equivalent expressions are often used to solve equations by simplifying both sides or isolating the variable. Here’s an example:

Solving Linear Equations

1. Simplify both sides of the equation:
3(x+2)=2x+6
Distribute on the left:
3x+6=2x+6
2. Combine like terms to isolate the variable:
3x+6−2x=6
x+6=6
3. Subtract 6 from both sides:
x=0

### Real-World Applications

Equivalent expressions are not just abstract algebraic concepts; they have practical applications in real-world scenarios. For example, they can be used in financial planning, engineering calculations, and solving problems involving rates and proportions.

Example in Financial Planning:

If you earn $15 per hour and work h hours, your total earnings can be represented by the expression 15h. If you receive a$30 bonus, the equivalent expression for your total earnings would be 15h+30.

### Examples of Equivalent Expressions

Example 1

Expression:
4(x+3)
Equivalent Expression:
4x+12

Explanation:
Using the distributive property, we multiply 4 by both x and 3:
4(x+3)=4⋅x+4⋅3=4x+12

Example 2

Expression:
5(a−2)

Equivalent Expression:
5a−10

Explanation:
Using the distributive property, we multiply 5 by both a and −2:
5(a−2)=5⋅a+5⋅(−2)=5a−10Example 3

Expression:
7+3(2y+4)
Equivalent Expression:
7+6y+12

Explanation:
First, use the distributive property to multiply 3 by 2y and 4:
3(2y+4)=6y+12
Then, add the constant 7:
7+6y+12=7+6y+12

Example 4

Expression:
2(3x+5)−x
Equivalent Expression:
6x+10−x
Simplified Equivalent Expression: 5x+10

Explanation:
Using the distributive property:
2(3x+5)=6x+10
Then, subtract x: 6x+10−x=5x+10

Example 5

Expression: (x+2)(x−3)
Equivalent Expression:
x²−x−6

Explanation:

Using the FOIL method (First, Outside, Inside, Last):
(x+2)(x−3)=x⋅x+x⋅(−3)+2⋅x+2⋅(−3)=x²−3x+2x−6=x²−x−6.

### Practice Questions

Question 1

Which of the following is equivalent to 3(2y+4)?

A. 6y+4
B. 6y+12
C. 6+4y
D. 3y+8

Explanation: Use the distributive property:
3(2y+4)=3⋅2y+3⋅4=6y+12.

Question 2

Which expression is equivalent to 4(a−5)+3a?

A. 4a−20+3a
B. 7a−20
C. 4a−5+3a
D. 7a−5

Explanation: First, use the distributive property:
4(a−5)=4a−20
4a−20+3a=7a−20.

Question 3

Simplify the expression 5(x+2)−3(x−1). Which of the following is equivalent?

A. 2x+13
B. 5x+10−3x+3
C. 2x+7
D. 8x+3

Explanation:

First, use the distributive property:
5(x+2)=5x+10
−3(x−1)=−3x+3
Combine the expressions:
5x+10−3x+3
(5x−3x)+(10+3)
2x+13.