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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).

### 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:
1. Distribute any coefficients: a, left parenthesis, b, x, plus minus, c, right parenthesis, equals, a, b, x, plus minus, a, c.
2. Combine any like terms on each side of the equation: x-terms with x-terms and constants with constants.
3. Arrange the terms in the same order, usually x-term before constants.
4. 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, plus, b, equals, c, x, plus, d for all values of x, then:
• a must equal c.
• b must equal d.
To find the value of unknown coefficients:
1. Distribute any coefficients on each side of the equation.
2. Combine any like terms on each side of the equation.
3. Set the coefficients on each side of the equation equal to each other.
4. 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, equals, 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, equals, start fraction, A, divided by, w, end fraction.
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.

TRY: IDENTIFYING EQUIVALENT EXPRESSIONS
Which of the following expressions is equivalent to 4, x, minus, 3 for all values of x ?

TRY: IDENTIFYING EQUIVALENT EXPRESSIONS
Which of the following expressions are equivalent to 7, x, plus, 1 ?

TRY: CALCULATING THE UNKNOWN COEFFICIENT
If start fraction, 1, divided by, 2, end fraction, left parenthesis, 6, x, plus, 8, right parenthesis, equals, k, x, plus, 4, what is the value of k ?
k, equals

TRY: REARRANGING THE FORMULA
If p, equals, m, v, which of the following correctly shows v in terms of p and m ?

## 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:
1. Distribute any coefficients: a, left parenthesis, b, x, plus minus, c, right parenthesis, equals, a, b, x, plus minus, a, c
2. Combine any like terms on each side of the equation: x-terms with x-terms and constants with constants
3. Arrange the terms in the same order, usually x-term before constants.
4. If all of the terms in the two expressions are identical, then the two expressions are equivalent.
To find the value of unknown coefficients:
1. Distribute any coefficients on each side of the equation.
2. Combine any like terms on each side of the equation.
3. Set the coefficients on each side of the equation equal to each other.
4. 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.