Expressions with One or More Variables

Evaluate expressions given values for 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 \begin{align*}10h + 25\end{align*}, 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 \begin{align*}x = 12\end{align*}. Find the value of \begin{align*}2x - 7\end{align*}.

Solution:

To find the solution, we substitute 12 for \begin{align*}x\end{align*} in the given expression. Every time we see \begin{align*}x\end{align*}, we replace it with 12.

\begin{align*}2x - 7 &= 2(12) - 7\\ &= 24 - 7\\ &= 17\end{align*}

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 \begin{align*}2x\end{align*} into 212 instead of 2 times 12!)

Example B

Let \begin{align*}y = -2. \end{align*} Find the value of \begin{align*} \frac {7} {y} - 11 y + 2 \end{align*}.

Solution

\begin{align*}\frac {7} {(-2)} - 11( -2 ) + 2 &= -3 \frac { 1 } { 2 } + 22 + 2\\ &= 24 - 3 \frac { 1 } { 2 }\\ &= 20 \frac { 1 } { 2 }\end{align*}

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 \begin{align*}(l)\end{align*} and width \begin{align*}(w)\end{align*}. 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 \begin{align*} A = \frac{ h } { 2 } (a + b)\end{align*}. Find the area of a trapezoid with bases \begin{align*}a = 10 \ cm\end{align*} and \begin{align*}b = 15 \ cm\end{align*} and height \begin{align*}h = 8 \ cm\end{align*}.

Solution:

To find the solution to this problem, we simply take the values given for the variables \begin{align*}a, \ b,\end{align*} and \begin{align*}h\end{align*}, and plug them in to the expression for \begin{align*}A\end{align*}:

\begin{align*}& 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\end{align*}

The area of the trapezoid is 100 square centimeters.

Watch this video for help with the Examples above.

Vocabulary

• When 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. We substitute the value in for the variable and simplify the expression.

Guided Practice

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

Solution

\begin{align*}3xy + \frac{6}{y}-2x &= 3(3)(-2) + \frac{6}{-2}-2(3)\\ &= -18-3-6)\\ &= -27\end{align*}

Practice

Evaluate 1-8 using \begin{align*}a = -3, \ b = 2, \ c = 5,\end{align*} and \begin{align*}d = -4\end{align*}.

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

For 9-11, the weekly cost \begin{align*}C\end{align*} of manufacturing \begin{align*}x\end{align*} remote controls is given by the formula \begin{align*}C = 2000 + 3x\end{align*}, 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?

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