# Expressions with One or More Variables

## Evaluate expressions given values for variables.

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Expressions with One or More Variables

### Multiple Variable Expressions

When given an algebraic expression, one of the most common things to do with it is evaluate it for some given value of the variable.

Take a look at this example to see how this works:

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

To find the solution, substitute 12 in place of \begin{align*}x\end{align*} in the given expression.

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

Note: In the first step of the problem, keep the substituted value in parentheses. This makes the written-out problem easier to follow, and helps 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!")

#### Evaluating an Expression

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

\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, \begin{align*}P=2l+2w\end{align*}, has two variables: length \begin{align*}(l)\end{align*} and width \begin{align*}(w).\end{align*} Be careful to substitute the appropriate value in the appropriate place.

#### Evaluating an Expression with Multiple Variables

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*}.

To find the solution to this problem, substitute the given values for variables \begin{align*}a, \ b,\end{align*} and \begin{align*}h\end{align*} in place of the appropriate letters in the equation.

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

### Example

#### Example 1

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*}.

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

### Review

Evaluate questions 1 through 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 questions 9 through 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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### Vocabulary Language: English

TermDefinition
algebraic The word algebraic indicates that a given expression or equation includes variables.
Algebraic Expression An expression that has numbers, operations and variables, but no equals sign.
Evaluate To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
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 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 "(" and ")" are used in algebraic expressions as grouping symbols.
substitute In algebra, to substitute means to replace a variable or term with a specific value.
Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.