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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?

Expressions with One or More Variables

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

Now try out your new vocabulary.

Let's evaluate \begin{align*}7y-11\end{align*}, when \begin{align*}y = 4\end{align*}:

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{Substitute the number 4 for}\ y.\\ &28 - 11 && \text{Multiply}\ 7 \ \text{by}\ 4.\\ &17 && \text{Subtract}\ 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.

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

\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*}.

Apply expressions to a real life situation:

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?

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.




Example 1

Earlier, you were asked how to find the area of a circle with a radius of 25 inches if you know the area of a general circle is approximately \begin{align*}3.14r^{2}\end{align*} with \begin{align*}r\end{align*} being the radius of the circle.

\begin{align*}3.14r^2=3.14(25)^2= 3.14(625)=1962.5\end{align*} 

The area of the circle is approximately 1962.5 in².

Example 2

 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*}.

\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*}:


Example 3

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

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.


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.
  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.

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.2. 

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


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.


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 "(" and ")" are used in algebraic expressions as grouping symbols.

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