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Expressions with One or More Variables

Evaluate expressions given values for variables.

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Practice Expressions with One or More Variables
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Expressions with One or More Variables

What if the paycheck for your summer job were represented by the algebraic expression $10h + 25$ , where h is the number of hours you work? If you worked 20 hours last week, how could you find the value of your paycheck? After completing this Concept, you'll be able to evaluate algebraic expressions like this one.

Guidance

When we are given an algebraic expression, one of the most common things we might have to do with it is evaluate it for some given value of the variable. The following example illustrates this process.

Example A

Let $x = 12$ . Find the value of $2x - 7$ .

Solution:

To find the solution, we substitute 12 for $x$ in the given expression. Every time we see $x$ , we replace it with 12.

$$2x - 7 &= 2(12) - 7\\ &= 24 - 7\\ &= 17$$

Note: At this stage of the problem, we place the substituted value in parentheses. We do this to make the written-out problem easier to follow, and to avoid mistakes. (If we didn’t use parentheses and also forgot to add a multiplication sign, we would end up turning $2x$ into 212 instead of 2 times 12!)

Example B

Let $y = -2.$ Find the value of $\frac {7} {y} - 11 y + 2$ .

Solution

$$\frac {7} {(-2)} - 11( -2 ) + 2 &= -3 \frac { 1 } { 2 } + 22 + 2\\ &= 24 - 3 \frac { 1 } { 2 }\\ &= 20 \frac { 1 } { 2 }$$

Many expressions have more than one variable in them. For example, the formula for the perimeter of a rectangle in the introduction has two variables: length $(l)$ and width $(w)$ . In these cases, be careful to substitute the appropriate value in the appropriate place.

Example C

The area of a trapezoid is given by the equation $A = \frac{ h } { 2 } (a + b)$ . Find the area of a trapezoid with bases $a = 10 \ cm$ and $b = 15 \ cm$ and height $h = 8 \ cm$ .

Solution:

To find the solution to this problem, we simply take the values given for the variables $a, \ b,$ and $h$ , and plug them in to the expression for $A$ :

$$& A = \frac { h } { 2 }(a + b) \qquad \text{Substitute} \ 10 \ \text{for} \ a, \ 15 \ \text{for} \ b, \ \text{and} \ 8 \ \text{for} \ h.\\ & A = \frac { 8 } { 2 }(10 + 15) \quad \text{Evaluate piece by piece.} \ 10 + 15 = 25; \ \frac { 8 } { 2 } = 4 .\\ & A = 4(25) = 100$$

The area of the trapezoid is 100 square centimeters.

Watch this video for help with the Examples above.

Vocabulary

• evaluate : To evaluate an algebraic expression for a given variable means to substitute a given value in place of the variable wherever it appears in the expression and then simplify.

Guided Practice

Let $x= 3$ and $y = -2.$ Find the value of $3xy + \frac{6}{y}-2x$ .

Solution

$$3xy + \frac{6}{y}-2x &= 3(3)(-2) + \frac{6}{-2}-2(3)\\ &= -18-3-6)\\ &= -27$$

Explore More

Evaluate 1-8 using $a = -3, \ b = 2, \ c = 5,$ and $d = -4$ .

1. $2a + 3b$
2. $4c + d$
3. $5ac - 2b$
4. $\frac { 2a } { c - d }$
5. $\frac { 3b } { d }$
6. $\frac { a - 4b } { 3c + 2d }$
7. $\frac { 1 } { a + b }$
8. $\frac { ab } {cd }$

For 9-11, the weekly cost $C$ of manufacturing $x$ remote controls is given by the formula $C = 2000 + 3x$ , where the cost is given in dollars.

1. What is the cost of producing 1000 remote controls?
2. What is the cost of producing 2000 remote controls?
3. What is the cost of producing 2500 remote controls?

Vocabulary Language: English

algebraic

algebraic

The word algebraic indicates that a given expression or equation includes variables.
Algebraic Expression

Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.
Evaluate

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Exponent

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Expression

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Order of Operations

Order of Operations

The order of operations specifies the order in which to perform each of multiple operations in an expression or equation. The order of operations is: P - parentheses, E - exponents, M/D - multiplication and division in order from left to right, A/S - addition and subtraction in order from left to right.
Parentheses

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
substitute

substitute

In algebra, to substitute means to replace a variable or term with a specific value.
Trapezoid

Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.