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Expressions with One or More Variables

Evaluate expressions given values for variables.

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Expressions with One or More Variables

Suppose you know the area of a circle is approximately \begin{align*}3.14r^{2}\end{align*}, where \begin{align*}r\end{align*} is the radius of the circle. What if a circle has a radius of 25 inches? How would you find its area? In this Concept, you'll learn how to substitute 25 inches into the expression, in place of the \begin{align*}r\end{align*} (radius), then evaluate the expression.

Guidance

In Algebra, evaluating an expression commonly means to replace any variables (letters) in the expression with given values, and then simplify the expression by performing any operations involved.

Example A

Evaluate \begin{align*}7y-11\end{align*}, when \begin{align*}y = 4\end{align*}.

Solution: Evaluate means to follow the directions, which is to take 7 times \begin{align*}y\end{align*} and subtract 11. Because \begin{align*}y\end{align*} is the number 4, we can evaluate our expression as follows:

\begin{align*}&7 \times (y) -11 && \text{First, substitute the number 4 in place of}\ y.\\ &7 \times 4 - 11 && \text{Then multiply}\ 7 \ \text{by}\ 4.\\ &28 - 11 && \text{Subtract}\ 11 \ \text{from}\ 28.\\ &17 && \text{The solution is}\ 17.\end{align*}

Because algebra uses variables to represent unknown quantities, the multiplication symbol \begin{align*}\times\end{align*} is often confused with the variable \begin{align*}x\end{align*}. To help avoid confusion, mathematicians replace the multiplication symbol with parentheses ( ) or the multiplication dot \begin{align*}\cdot\end{align*} , or by eliminating the symbol entirely if a number is being multiplied by a variable. For example, all four of these expressions mean the same thing:

\begin{align*}4 \times a + 3 \times b\\ 4(a)+3(b)\\ 4 \cdot a + 3 \cdot b\\ 4a+3b\end{align*}

Example B

Rewrite \begin{align*}P = 2 \times l + 2 \times w\end{align*} with alternative multiplication symbols.

Solution: \begin{align*}P = 2 \times l + 2 \times w\end{align*} can be written as \begin{align*}P = 2 \cdot l + 2 \cdot w\end{align*}.

It can also be written as \begin{align*}P = 2l + 2w\end{align*}.

The following is a real-life example that shows the importance of evaluating a mathematical variable.

Example C

To prevent major accidents or injuries, horses are to be fenced in a rectangular pasture. If the dimensions of the pasture are 300 feet by 225 feet, how much fencing should the ranch hand purchase to enclose the pasture?

Solution: Begin by drawing a diagram of the pasture and labeling what you know.

To find the amount of fencing needed, you must add all the sides together:

\begin{align*}L + L + W + W\end{align*}

By substituting the dimensions of the pasture for the variables \begin{align*}L\end{align*} and \begin{align*}W\end{align*}, the expression becomes:

\begin{align*}300 + 300 + 225 + 225\end{align*}

Now evaluate by adding the values together. The ranch hand must purchase \begin{align*}300 + 300 + 225 + 225 = 1,050\end{align*} feet of fencing.

Guided Practice

1. Write the expression \begin{align*}2\times a\end{align*} in a more condensed form and then evaluate it for \begin{align*}3=a\end{align*}.

2. If it costs $9.25 for each movie ticket, how much does it cost for 4 people to see a movie?

Solutions:

1. \begin{align*}2\times a\end{align*} can be written as \begin{align*}2(a).\end{align*} We can substitute 3 for \begin{align*}a\end{align*}: 2(3) = 6.

2. Since each movie ticket is $9.25, we multiply this price by the 4 people buying tickets to get the total cost:

\begin{align*}\$9.25\times 4=\$37.00\end{align*}

It costs $37 for 4 people to see a movie.

Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Variable Expressions (12:26)

In 1–4, write the expression in a more condensed form by leaving out the multiplication symbol.

  1. \begin{align*}2 \times 11x\end{align*}
  2. \begin{align*}1.35 \cdot y\end{align*}
  3. \begin{align*}3 \times \frac{1}{4}\end{align*}
  4. \begin{align*}\frac{1}{4} \cdot z\end{align*}

In 5–9, evaluate the expression.

  1. \begin{align*}5m + 7\end{align*} when \begin{align*}m = 3\end{align*}
  2. \begin{align*}\frac{1}{3} (c)\end{align*} when \begin{align*}c = 63\end{align*}
  3. $8.15(\begin{align*}h\end{align*}) when \begin{align*}h = 40\end{align*}
  4. \begin{align*}(k-11) \div 8\end{align*} when \begin{align*}k = 43\end{align*}
  5. \begin{align*}(-2)^2 + 3(j)\end{align*} when \begin{align*}j = -3\end{align*}

In 10–17, evaluate the expression. Let \begin{align*}a = \text{-}3, \ b = 2, \ c = 5, \text{ and }d= \text{-} 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*}

In 18–25, evaluate the expression. Let \begin{align*}x = \text{-}1, \ y = 2, \ z = \text{-}3, \text{ and }w=4.\end{align*}

  1. \begin{align*}8x^3 \end{align*}
  2. \begin{align*}\frac{5x^2}{6z^3}\end{align*}
  3. \begin{align*}3z^2 - 5w^2\end{align*}
  4. \begin{align*}x^2 - y^2\end{align*}
  5. \begin{align*}\frac{z^3 + w^3}{z^3 - w^3}\end{align*}
  6. \begin{align*}2x^2 - 3x^2 + 5x - 4\end{align*}
  7. \begin{align*}4w^3 + 3w^2 - w + 2\end{align*}
  8. \begin{align*}3 + \frac{1}{z^2}\end{align*}

In 26–30, evaluate the expression in each real-life problem.

  1. The measurement around the widest part of these holiday bulbs is called their circumference. The formula for circumference is \begin{align*}2(r) \pi\end{align*}, where \begin{align*}\pi \approx 3.14\end{align*} and \begin{align*}r\end{align*} is the radius of the circle. Suppose the radius is 1.25 inches. Find the circumference.
  2. The dimensions of a piece of notebook paper are 8.5 inches by 11 inches. Evaluate the area of the paper. The formula for area is length \begin{align*}\times\end{align*} width.
  3. Sonya purchased 16 cans of soda at $0.99 each. What is the amount Sonya spent on soda?
  4. Mia works at a job earning $4.75 per hour. How many hours should she work to earn $124.00?
  5. The area of a square is the side length squared. Evaluate the area of a square with a side length of 10.5 miles.

Answers for Explore More Problems

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

Vocabulary

substitute

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

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.

Exponent

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

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.

Image Attributions

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