1.1: Variable Expressions
Who Speaks Math, Anyway?
When someone is having trouble with algebra, they may say, “I don’t speak math!” While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a secondary language that you must learn in order to be successful. There are verbs and nouns in math, just like in any other language. In order to understand math, you must practice the language.
A verb is a “doing” word, such as running, jumps, or drives. In mathematics, verbs are also “doing” words. A math verb is called an operation. Operations can be something you have used before, such as addition, multiplication, subtraction, or division. They can also be much more complex like an exponent or square root.
Example: Suppose you have a job earning $8.15 per hour. What could you use to quickly find out how much money you would earn for different hours of work?
Solution: You could make a list of all the possible hours, but that would take forever! So instead, you let the “hours you work” be replaced with a symbol, like \begin{align*}h\end{align*} for hours, and write an equation such as:
\begin{align*}amount \ of \ money = 8.15(h)\end{align*}
A noun is usually described as a person, place, or thing. In mathematics, nouns are called numbers and variables. A variable is a symbol, usually an English letter, written to replace an unknown or changing quantity.
Example: What variables would be choices for the following situations?
a. the number of cars on a road
b. time in minutes of a ball bounce
c. distance from an object
Solution: There are many options, but here are a few to think about.
a. Cars is the changing value, so \begin{align*}c\end{align*} is a good choice.
b. Time is the changing value, so \begin{align*}t\end{align*} is a good choice.
c. Distance is the varying quantity, so \begin{align*}d\end{align*} is a good choice.
Why Do They Do That?
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: 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,
\begin{align*}&7 \times 4 - 11 && \text{We have “substituted” the number 4 for}\ y.\\ &28 - 11 && \text{Multiplying}\ 7 \ \text{and}\ 4\\ &17 && \text{Subtracting}\ 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 ( ), the multiplication dot \begin{align*}\cdot\end{align*}, or by writing the expressions side by side.
Example: 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: 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 1,050 feet of fencing.
Practice Set
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 a multiplication symbol.
- \begin{align*}2 \times 11x\end{align*}
- \begin{align*}1.35 \cdot y\end{align*}
- \begin{align*}3 \times \frac{1}{4}\end{align*}
- \begin{align*}\frac{1}{4} \cdot z\end{align*}
In 5 – 9, evaluate the expression.
- \begin{align*}5m + 7\end{align*} when \begin{align*}m = 3\end{align*}.
- \begin{align*}\frac{1}{3} (c)\end{align*} when \begin{align*}c = 63\end{align*}.
- $8.15(\begin{align*}h\end{align*}) when \begin{align*}h = 40\end{align*}.
- \begin{align*}(k-11) \div 8\end{align*} when \begin{align*}k = 43\end{align*}.
- Evaluate \begin{align*}(-2)^2 + 3(j)\end{align*} when \begin{align*}j = -3\end{align*}.
In 10 – 17, evaluate the expressions. Let \begin{align*}a = -3, \ b = 2, \ c = 5,\end{align*} and \begin{align*}d = -4\end{align*}.
- \begin{align*}2a + 3b\end{align*}
- \begin{align*}4c + d\end{align*}
- \begin{align*}5ac - 2b\end{align*}
- \begin{align*}\frac{2a}{c - d}\end{align*}
- \begin{align*}\frac{3b}{d}\end{align*}
- \begin{align*}\frac{a - 4b}{3c + 2d}\end{align*}
- \begin{align*}\frac{1}{a + b}\end{align*}
- \begin{align*}\frac{ab}{cd}\end{align*}
In 18 – 25, evaluate the expressions. Let \begin{align*}x = -1, \ y = 2, \ z = -3,\end{align*} and \begin{align*}w=4\end{align*}.
- \begin{align*}8x^3 \end{align*}
- \begin{align*}\frac{5x^2}{6z^3}\end{align*}
- \begin{align*}3z^2 - 5w^2\end{align*}
- \begin{align*}x^2 - y^2\end{align*}
- \begin{align*}\frac{z^3 + w^3}{z^3 - w^3}\end{align*}
- \begin{align*}2x^2 - 3x^2 + 5x - 4\end{align*}
- \begin{align*}4w^3 + 3w^2 - w + 2\end{align*}
- \begin{align*}3 + \frac{1}{z^2}\end{align*}
In 26 – 30, choose an appropriate variable to describe each situation.
- The number of hours you work in a week
- The distance you travel
- The height of an object over time
- The area of a square
- The number of steps you take in a minute
In 31 – 35, underline the math verb(s) in the sentence.
- The product of six and \begin{align*}v\end{align*}
- Four plus \begin{align*}y\end{align*} minus six
- Sixteen squared
- \begin{align*}U\end{align*} divided by 3 minus eight
- The square root of 225
In 36 – 40, evaluate the real-life problems.
- 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
- 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.
- Sonya purchases 16 cans of soda at $0.99 each. What is the amount Sonya spent on soda?
- Mia works at a job earning $4.75 per hour. How many hours should she work to earn $124.00?
- The area of a square is the side length squared. Evaluate the area of a square with side length 10.5 miles.