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1.2: Expressions with One or More Variables

Difficulty Level: Advanced Created by: CK-12
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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 representing the area and evaluate the expression.


Just like in the English language, mathematics uses several words to describe one thing. For example, sum, addition, more than, and plus all mean to add numbers together. The following definition shows an example of this.

Definition: To evaluate means to follow the verbs in the math sentence. Evaluate can also be called simplify or answer.

To begin to evaluate a mathematical expression, you must first substitute a number for the variable.

Definition: To substitute means to replace the variable in the sentence with a value.

Now try out your new vocabulary.

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 4 - 11 && \text{We have “substituted” the number 4 for}\ y.\\ &28 - 11 && \text{We have multiplied}\ 7 \ \text{and}\ 4.\\ &17 && \text{We have subtracted}\ 11 \ \text{from}\ 28.\\ &\text{The solution is}\ 17.\end{align*}

Because algebra uses variables to represent the 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 writing the expressions side by side.

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

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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 a movie ticket, how much does it cost for 4 people to see a movie?


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


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.


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 = -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*}

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

    Christmas Baubles by Petr Kratochvil

  2. The dimensions of a piece of notebook paper are 8.5 inches by 11 inches. Evaluate the writing 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.

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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.

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Feb 24, 2012
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