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1.2: Expressions with One or More Variables

Difficulty Level: At Grade Created by: CK-12
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Suppose you had a summer job that paid you $25 per week, plus $10 per hour. Your weekly paycheck could be represented as \begin{align*}10h+25,\end{align*}10h+25,where \begin{align*}h\end{align*}his the number of hours you worked that week. Can you find the value of your paycheck for a week when you worked 20 hours? After completing this Concept, you'll be able to evaluate algebraic expressions like this one.

Watch This

CK-12 Foundation: 0102s evaluate algebraic expressions

Guidance

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:

Example A

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

Solution:

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

\begin{align} 2x - 7 & = 2(12) - 7\\ & = 24 - 7\\ & = 17 \end{align}2x7=2(12)7=247=17

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*}2x" into "212" instead of "2 times 12!")

Example B

Let \begin{align*}y = -2. \end{align*}y=2. Find the value of \begin{align*} \frac {7} {y} - 11 y + 2.\end{align*}7y11y+2.

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}7(2)11(2)+2=312+22+2=24312=2012

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*}P=2l+2w, has two variables: length \begin{align*}(l)\end{align*}(l) and width \begin{align*}(w).\end{align*}(w). 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*}A=h2(a+b). Find the area of a trapezoid with bases \begin{align*}a = 10 \ cm\end{align*}a=10 cm and \begin{align*}b = 15 \ cm\end{align*}b=15 cm and height \begin{align*}h = 8 \ cm\end{align*}h=8 cm.

Solution:

To find the solution to this problem, substitute the given values for variables \begin{align*}a, \ b,\end{align*}a, b, and \begin{align*}h\end{align*}h 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}AA=h2(a+b) Substitute 10 for a, 15 for b, and 8 for h.=(8)2((10)+(15))Evaluate piece by piece. 10+15=25 and 82=4.=4(25)=100

You may wish to watch this video for help with the examples above.

CK-12 Foundation: Evaluate Algebraic Expressions

Guided Practice

Let \begin{align*}x= 3\end{align*}x=3 and \begin{align*}y = -2. \end{align*}y=2. Find the value of \begin{align*} 3xy + \frac{6}{y}-2x \end{align*}3xy+6y2x.

Solution

\begin{align} 3xy + \frac{6}{y}-2x & = 3(3)(-2) + \frac{6}{(-2)}-2(3)\\ & = (-18)+(-3)-(6)\\ & = -27 \end{align}3xy+6y2x=3(3)(2)+6(2)2(3)=(18)+(3)(6)=27

Explore More

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

  1. \begin{align*}2a + 3b\end{align*}2a+3b
  2. \begin{align*}4c + d\end{align*}4c+d
  3. \begin{align*}5ac - 2b\end{align*}5ac2b
  4. \begin{align*} \frac { 2a } { c - d }\end{align*}2acd
  5. \begin{align*} \frac { 3b } { d }\end{align*}3bd
  6. \begin{align*} \frac { a - 4b } { 3c + 2d }\end{align*}a4b3c+2d
  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?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.2. 

Vocabulary

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.

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Sep 26, 2012
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