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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).

- 3 + 2

and

2 - 3

are equivalent operations. The numbers are presented in different orders, but both are equal to

- 1

\frac{1}{2}

and

\frac{5}{10}

are equivalent fractions. The numbers are different, but both are equal to

0.5

2 (x + 1)

and

2 x + 2

are equivalent algebraic expressions. The value of the expression are the same for all values of

x

$x$ ‍	$2 (x + 1)$ ‍	$2 x + 2$ ‍
$0$ ‍	$2 (0 + 1) = 2$ ‍	$2 (0) + 2 = 2$ ‍
$1$ ‍	$2 (1 + 1) = 4$ ‍	$2 (1) + 2 = 4$ ‍
$2$ ‍	$2 (2 + 1) = 6$ ‍	$2 (2) + 2 = 6$ ‍
$3$ ‍	$2 (3 + 1) = 8$ ‍	$2 (3) + 2 = 8$ ‍

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 $2$ ‍ or more variables

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: $a (b x \pm c) = a b x \pm a c$ ‍.
Combine any like terms on each side of the equation: $x$ ‍-terms with $x$ ‍-terms and constants with constants.
Arrange the terms in the same order, usually $x$ ‍-term before constants.
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 must be equivalent. For example, if

a x + b = c x + d

for all values of

x

, then:

$a$ ‍ must equal $c$ ‍.
$b$ ‍ must equal $d$ ‍.

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

A = l w

. 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 = \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.

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: $a (b x \pm c) = a b x \pm a c$ ‍
Combine any like terms on each side of the equation: $x$ ‍-terms with $x$ ‍-terms and constants with constants
Arrange the terms in the same order, usually $x$ ‍-term before constants.
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.

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ljone004
Posted 4 years ago. Direct link to ljone004's post “Write an equivalent multi...”
Write an equivalent multiplication expression with two factors
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sarika.singh
Posted 3 years ago. Direct link to sarika.singh's post “expressions are equivalen...”
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
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Becca Lillie
Posted a year ago. Direct link to Becca Lillie's post “I would love some visuals...”
I would love some visuals or more examples to help me understand this better. I haven’t taken algebra since the 2010s
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Haydee Marquez
Posted 3 months ago. Direct link to Haydee Marquez's post “some are easy, but the km...”
some are easy, but the km one is hard and I don't understand.
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lizzy.kobelamoloisi
Posted 2 years ago. Direct link to lizzy.kobelamoloisi's post “I don't understand the Qu...”
I don't understand the Question
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Aaron Aparicio Chagolla :>
Posted 8 months ago. Direct link to Aaron Aparicio Chagolla :>'s post “My icon/profile pic is di...”
My icon/profile pic is different.
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janneylloyd460
Posted 2 years ago. Direct link to janneylloyd460's post “How can you tell the diff...”
How can you tell the difference in which expression or formula you used based on the equation that been presented
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kduy9180
Posted 8 months ago. Direct link to kduy9180's post “Is \frac{\frac{x-3}{x-4}-...”
Is \frac{\frac{x-3}{x-4}-1}{\frac{x-3}{x-4}-2} equivalent to \frac{1}{\:5-x}?
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Paolo
Posted a month ago. Direct link to Paolo's post “i dont understand any of ...”
i dont understand any of the questions
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(2 votes)
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-_-
Posted a year ago. Direct link to -_-'s post “expressions are equivalen...”
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
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer

Course: Praxis Core Math > Unit 1