Main content

### Course: Praxis Core Math > Unit 1

Lesson 4: Algebra- Algebraic properties | Lesson
- Algebraic properties | Worked example
- Solution procedures | Lesson
- Solution procedures | Worked example
- Equivalent expressions | Lesson
- Equivalent expressions | Worked example
- Creating expressions and equations | Lesson
- Creating expressions and equations | Worked example
- Algebraic word problems | Lesson
- Algebraic word problems | Worked example
- Linear equations | Lesson
- Linear equations | Worked example
- Quadratic equations | Lesson
- Quadratic equations | Worked example

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Equivalent expressions | Lesson

## What are equivalent expressions?

**Equivalent expressions**are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).

### What skills are tested?

- Distributing coefficients and combining like terms in algebraic expressions
- Recognizing equivalent algebraic expressions
- Solving for an unknown coefficient using two equivalent expressions
- Rearranging formulas containing
or more variables$2$

## How do we recognize equivalent expressions?

Questions about equivalent expressions usually feature both and . To check which complex expression is equivalent to the simple expression:

- Distribute any coefficients:
.$a(bx\pm c)=abx\pm ac$ - Combine any like terms on each side of the equation:
-terms with$x$ -terms and constants with constants.$x$ - Arrange the terms in the same order, usually
-term before constants.$x$ - If all of the terms in the two expressions are identical, then the two expressions are equivalent.

## How do we solve for unknown coefficients?

Some questions will present us with an equation with algebraic expressions on both sides. On one side, there will be an unknown coeffient, and the question will ask us to find its value.

For the equation to be true for all values of the variable, the two expressions on each side of the equation $ax+b=cx+d$ for all values of $x$ , then:

*must*be equivalent. For example, if must equal$a$ .$c$ must equal$b$ .$d$

To find the value of unknown coefficients:

- Distribute any coefficients on each side of the equation.
- Combine any like terms on each side of the equation.
- Set the coefficients on each side of the equation equal to each other.
- Solve for the unknown coefficient.

## How do we rearrange formulas?

Formulas are equations that contain $2$ or more variables; they describe relationships and help us solve problems in geometry, physics, etc.

Since a formula contains multiple variables, sometimes we're interested in writing a specific variable in terms of the others. For example, the formula for the area, $A$ , for a rectangle with length $l$ and width $w$ is $A=lw$ . It's easy to calculate $A$ using the formula if we know $l$ and $w$ . However, if we know $A$ and $w$ and want to calculate $l$ , the formula that best helps us with that is an equation in which $l$ is in terms of $A$ and $w$ , or $l={\displaystyle \frac{A}{w}}$ .

Just as we can add, subtract, multiply, and divide constants, we can do so with variables. To isolate a specific variable, perform the same operations on both sides of the equation until the variable is isolated. The new equation is equivalent to the original equation.

## Your turn!

## Things to remember

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.

To check whether a more complex expression is equivalent to a simpler expression:

- Distribute any coefficients:
$a(bx\pm c)=abx\pm ac$ - Combine any like terms on each side of the equation:
-terms with$x$ -terms and constants with constants$x$ - Arrange the terms in the same order, usually
-term before constants.$x$ - If all of the terms in the two expressions are identical, then the two expressions are equivalent.

To find the value of unknown coefficients:

- Distribute any coefficients on each side of the equation.
- Combine any like terms on each side of the equation.
- Set the coefficients on each side of the equation equal to each other.
- Solve for the unknown coefficient.

To isolate a specific variable in a formula, perform the same operations on both sides of the equation until the variable is isolated.

## Want to join the conversation?

- expressions are equivalent to

−

-8/11

−

-3/4

−

-1/4

−

-11/8

−

-4/3

−

-4/1

minus, start fraction, 8, divided by, 11, end fraction, minus, start fraction, 3, divided by, 4, end fraction, minus, start fraction, 1, divided by, 4, end fraction(3 votes) - I would love some visuals or more examples to help me understand this better. I haven’t taken algebra since the 2010s(2 votes)
- Write an equivalent multiplication expression with two factors(2 votes)
- How can you tell the difference in which expression or formula you used based on the equation that been presented(2 votes)
- some are easy, but the km one is hard and I don't understand.(1 vote)
- I don't understand the Question(1 vote)
- My icon/profile pic is different.(1 vote)
- Is \frac{\frac{x-3}{x-4}-1}{\frac{x-3}{x-4}-2} equivalent to \frac{1}{\:5-x}?(1 vote)
- i dont understand any of the questions(1 vote)
- expressions are equivalent to

−

-8/11

−

-3/4

−

-1/4

−

-11/8

−

-4/3

−

-4/1

minus, start fraction, 8, divided by, 11, end fraction, minus, start fraction, 3, divided by, 4, end fraction, minus, start fraction, 1, divided by, 4, end fraction(0 votes)