5-1: Variables and Algebraic Expressions

Hazel McKenna; Leo Chang; Roxanne Brinkerhoff

CHAPTER 5: LINEAR EQUATIONS IN ONE VARIABLE

5-1: Variables and Algebraic Expressions

Constants and Variables

The definition of the word constant is never changing; stays the same. In mathematics, a constant is a number; the value of the number never changes. But there are very few things in life that remain constant, so we need a way of describing a value that can change or vary. In mathematics, we use a symbol or a letter to stand in for a quantity that either varies (i.e., it is not constant) or whose value we do not know. Such a symbol is called a variable.

In 1908, German mathematician Felix Klein wrote,^[1]

“one may well declare that

real mathematics begins with operations with letters.”

and Polish-American mathematician Alfred Tarski wrote in 1941 that,^[2]

“the invention of variables constitutes a turning point

in the history of mathematics.”

Variables are so important in math as they allow mathematical ideas to be communicated clearly and concisely. Variables also make mathematics more generally applicable. For example, to determine the area of a flower bed that measures 5 feet by 2 feet, we can multiply the length by the width to get $5\;\text{ft}\times2\;\text{ft}=10\; \text{ft}^2$ . But that multiplication only works when the length is 5 feet and the width is 2 feet. To express the fact that the area of a rectangular two-dimensional space is always found by multiplying its length by its width, we use variables to create a formula: $A = l \times w$ , where $A$ represents area, $l$ represents length and $w$ represents width. This formula can be used to determine the area of any rectangle provided we know the values for the length and the width.

Another benefit is that we may use variables to represent mathematical relations, which makes it easier when solving problems. For example, we may use the letter $p$ to represent the original price of a pair of shoes that we are asked to determine given that they are on sale for $50 after a 20% discount. The quantitative relation can be written in words as,

the original price – the discount = the sale price

which, after defining the variable $p$ to represent the unknown original price, can be written symbolically as

$p-20\%\times p=50$

After performing some algebraic manipulation on the equation, we may then determine the original price $p$ .

Variables, Terms, and Expressions

Explore 1 – Variables

Suppose you buy three combo meals from a fast-food restaurant, where the price of each combo meal is $10.99. Determine the cost of the three meals.

Solution

We can determine the cost in two ways using either multiplication or addition:

multiplication: cost = 3 $\times$ $10.99 = $32.97

addition: cost = $10.99 + $10.99 + $10.99 = $32.97

Note that the cost is the same whether we use multiplication or addition.

Suppose you do not know the cost of a combo meal, so you define the variable $x$ to represent the price of a combo meal. Determine the cost of three meals using multiplication and then using addition. Draw a conclusion from your efforts.

Solution

multiplication: cost = $3\times x=3x$

addition: cost = $x + x + x$

Since the cost must be the same using multiplication or addition, we can conclude that $x + x + x=3x$ .

Explore 2 – Addition of variables

A constant multiplied by a variable is equivalent to that variable being added multiple times. For example, $3x=x + x + x$ .

Simplify $3x+4x$

Solution

Since $3x=x+x+x$ and $4x=x+x+x+x$ , $3x+4x=(x+x+x)+(x+x+x+x)=7x$

Simplify $5x+3x$

Solution

Since $5x=x+x+x+x+x$ and $3x=x+x+x$ , $5x+3x=(x+x+x+x+x)+(x+x+x)=8x$

Write a rule that could be used to add $3x+4x$ or $5x+3x$

Solution

$3x+4x=(3+4)x=7x$ and $5x+3x=(5+3)x=8x$

Add the coefficients and keep the common variable.

Simplify $ax+bx$

Solution

Using the rule add the coefficients and keep the common variable: $ax+bx=(a+b)x$

When a variable is multiplied by a constant it forms a term. When terms have exactly the same variable(s) they are called like terms. Only like terms can be added and subtracted by adding or subtracting the coefficients and keeping the common variable.

Explore 3 – Expressions

Holly jumped on a dating app and matched with several people. She went on three dinner dates, five coffee dates, and two movie dates. Suppose the variables $x, \;y,$ and $z$ represent the cost of a dinner date, a coffee date, and a movie date, respectively. Use these variables to represent the total amount Holly spent on her dating experiences, if she paid.

Solution

Three dinner dates cost $3x$ ; five coffee dates cost $5y$ ; two movie dates cost $2z$ .

So, the total cost = $3x+5y+2z$ . These are not like terms so cannot be simplified.

When terms with variables are added or subtracted, they form an algebraic expression. Expressions can be simplified (e.g., by adding or subtracting like terms) or they can be evaluated.

Explore 4 – Simplifying and evaluating an expression

A grocery store manager has noticed that certain foods have the same price per item or per pound. The table shows the prices in terms of variables.

Grocery	Price
Apple	$ $x$ per pound
Cucumber	$ $x$ per item
Potato	$ $z$ per bag
Strawberry	$ $y$ per box
Eggs	$ $z$ per box
Soda	$ $y$ per 6-pack

Write an expression that represents the total cost of buying 5 pounds of apples, 2 boxes of strawberries, and 4 cucumbers. Simplify the expression.

Solution

cost of 5 pounds of apples = $5x$ ; cost of 2 boxes of strawberries = $2y$ ; cost of 4 cucumbers = $4x$

So, total cost = $5x+2y+4x=(5x+4x)+2y=9x+2y$ The two terms $5x$ and $4x$ are like terms and can be combined to $9x$ .

Suppose $x=\$1.29, \;y = \$1.99,$ and $z = \$2.99$ . Determine the total cost of buying 5 pounds of apples, 2 boxes of strawberries, and 4 cucumbers.

Solution

To evaluate an expression, we replace each variable with its given value then simplify:

total cost = $9x+2y=9(\$1.29)+2(\$1.99)=\$15.59$

Write an expression that represents the total cost of buying three 6 packs of soda, 4 boxes of strawberries, and 6 bags of potatoes. Simplify the expression.

Solution

cost of three 6-packs of soda = $3y$ ; cost of 4 boxes of strawberries = $4y$ ; cost of 6 bags of potatoes = $6z$

So, total cost = $3y+4y+6z=(3y+4y)+6z=7y+6z$ The two terms 3y and 4y are like terms and can be combined to $7y$ .

Suppose $x=\$1.29, \;y = \$1.99,$ and $z = \$2.99$ . Determine the total cost of buying three 6 packs of soda, 4 boxes of strawberries, and 6 bags of potatoes.

Solution

To evaluate an expression, we replace each variable with its given value then simplify:

total cost = $7y+6z=7(\$1.99)+6(\$2.99)=\$31.87$

Constants, Variables, Terms, and Expressions

A constant is a number that never changes value.

A variable is a symbol or letter that represents a quantity that either varies (i.e., it is not constant) or whose value we do not know.

A term is a constant, a variable, or the product or quotient of numbers and variables: $9x; \;8xy; \;-3x^2y;\;\dfrac{6}{x};\;\-\dfrac{2x}{5y^2z}$

A coefficient is the constant in a term: $-7x$ has a coefficient of –7

Like terms are terms that have exactly the same variables with exactly the same exponents: $5x$ and $-4x$ or $3x^2y$ and $8x^2y$

Only like terms can be added or subtracted.

An algebraic expression is a term or the combination of terms: $3x+8y-5z$ or $\dfrac{4x^2+8x-7}{3x-5}$

Explore 5 – Evaluating an expression

Weight balance is very important for an airplane to take off, land, and fly safely. Suppose the weight in the front of a plane is represented by the expression $50x + 8y + 20z$ and the weight in the rear is $20r + 6s + 30t$ . Are the weights balanced between the front and rear if $x = 20kg, \;y = 70kg, \;z = 40kg, \;r = 25kg, \;s = 200kg$ , and $t = 45kg$ ?

Solution

To evaluate the weight, we substitute each variable with its given value.

The weight in the front

$\begin{align*}&=50x + 8y + 20z\\&=50\times20kg+8\times70kg+20\times40kg\\&=1000kg+560kg+800kg\\&=2360kg\end{align*}$

.

The weight in the back

$\begin{align*}&=20r + 6s + 30t\\&=20\times25kg+6\times200kg+30\times45kg\\&=500kg+1200kg+1350kg\\&=3,050kg\end{align*}$

.

The weight of the back is 690kg more than the weight in the front. No, the weights are not balanced.

Reflect Icon

What is a variable? Why are variables used?

Show/Hide Answer

A variable is a value holder, which often exists in the form of a lower-case letter. When solving problems, a variable may be used to represent an unknown value, or as a changing amount in a formula.

What does it mean to “evaluate an expression”?

Show/Hide Answer

To evaluate an expression is to substitute variables in the expression with values/numbers such that the value of the expression with the given values is determined.

Examples

1. A cube is a three-dimensional object that is square on each side. The surface area of each side of a cube equals the square of the length of the side. i.e. if $x$ represents the length of each side, the surface area of any side is $x^2$ .

a) Write an expression that represents the surface area of the whole cube.

b) Determine the surface area of the cube when $x=3cm$ , $2\;ft$ , and $1.4\;in$

2. A scientist records a temperature of 50.4°C. Use the expression $1.8C+32$ to convert this temperature to degrees Fahrenheit.

Show/Hide Answer

a) $6x^2$ b) 54cm²; 24 ft²; 11.76 in²
122.72°F

Writing Algebraic Expressions

We may use variables to represent any unknown values in a problem, and then translate the mathematical relations in the problem into an algebraic expression. When we introduce a variable into the mix, we must define the variable. This means that we must give the variable a name and tell people what the variable represents.

Explore 6 – Translating to an algebraic expression

KoSze plays a game with his son. KoSze says, “I have a number. I multiply it by 5, then add 10, then take away three times the number, and then divide the result by 2.

Translate KoSze’s words into an algebraic expression, then simplify the expression.

Solution

Let $n=$ a number. This is defining the variable.

The number multiplied by 5 is equal to $5n$ . Then 10 is added to this:

$5n+10$

Three times the number is equal to $3n$ . “Take away” means “subtract”, so, now we have:

$5n+10-3n$

Finally we divide by 2:

$\dfrac{5n+10-3n}{2}$

The two terms $5n$ and $3n$ are like terms. Therefore, $5n-3n=2n$

This leaves us with,

$\dfrac{2n+10}{2}$

which can then be further simplified by distributing the 2 to both terms on the numerator:

$\dfrac{2n+10}{2}=\dfrac{2n}{2}+\dfrac{10}{2}=n+5$

Explore 7 – Writing an algebraic expression

Cathy rented a U-Haul truck to move from Denver to Salt Lake City. She was charged $29.99 per day and $0.99 per mile. She rented the truck for three days. As part of a promotion, she got the first 10 miles free.

Translate the problem into an algebraic expression to represent the total amount of the rental fee Cathy paid. Then simplify the expression.

Solution

The first part of the cost comes from the daily rental fee. Cathy rented the truck for three days and the daily rental charge was $29.99 per day. Consequently, the daily rental charge was $3\times\$29.99=\$89.97$ .

The second part of the cost comes from the mileage she drives. We need to define a variable: let $m=$ the number of miles driven.

Since the first 10 miles is free, the total number of miles that was charged is $m-10$ . Since the charge for mileage was $0.99 per mile, the total mileage charge was $\$0.99\times(m-10)$ .

Therefore, the total charge is the sum of the daily rental charge and the mileage charge, which is equal to:

$89.97+0.99 (m-10)$ .

We can simplify this by distributing the 0.99 to both the m and the –10:

$89.97+0.99m-0.99\times10=89.97-9.9+0.99m=80.07+0.99m$

So, the total charge is:

$\$(80.07+0.99m)$ .

Reflect Icon

What does it mean to translate a problem to an algebraic expression?

Show/Hide Answer

It means to turn the mathematical relations in a problem into a mathematical expression where the unknown values in the relations are represented by defined variables.

Practice Exercises

The CDC considers a person having a fever if the person’s body temperature at least 100.4°F. Use the expression $(F - 32)\div1.8$ to convert the temperature to degrees Celsius.
Bella wants to put a dog run at the side of her house. She knows she has enough room for the width to be 8 feet but she can’t remember the length. a) If she defines $l$ to be the length of the dog run, determine its perimeter in terms of $l$ . b) Calculate the perimeter of the dog run if $l=6\;ft,\;10\;ft,\;12\;ft$ .
Leo is buying snacks for a football party. He needs to buy 4 cases of beer, 3 bags of chips, 3 dozen hotdogs, 3 dozen hotdog buns, 2 bottles of salsa. Suppose the beer costs $ $x$ per case, chips cost $ $y$ per bag, hotdogs cost $ $y$ per dozen, hotdog buns cost $ $y$ per dozen, and salsa costs $ $y$ per bottle. a) Write an expression that represents the cost of the snacks, then simplify it. b) Use the simplified expression to determine the cost when $x=\$24.99,\;y=\$4.39,\;z=\$6.29$
Michael’s set monthly expenses are rent, electricity, gas, phone, Internet and car payment. His variable expenses are food, personal and household supplies, entertainment, and gas for his car. a) If his rent is $1800 per month, electricity bill is $58 per month, gas bill is $63 per month, phone is $65 per month, Internet is $75 per month, and car payment is $325 per month, what are his total fixed expenses? b) Define a variable for each of his variable expenses, then write an expression to show his total monthly expenses.
Diane is making naan flatbread. The recipe says to bake the bread at 180° Celsius. The temperature in ° Fahrenheit = $(\frac{9}{5}C+32$ , where $C$ = the temperature in ° Celsius. What temperature should she use to bake her bread?
Michelle is remodeling her basement and needs to borrow an orbital floor sander to level and etch her concrete floor. The cost is $56 per day. She also need a concrete etcher tool for the sander and that rents for $89 per day. a) If $d$ = number of rental days, write an expression to determine the total cost of renting the tools. b) If she rents the tools for 2 days, use your expression to determine the total cost.

Show/Hide Answer

38°C
a) $2l+16$ b) 28 ft; 26 ft; 40 ft
a) $4x+3y+3y+3y+2z=4x+9y+2z$ b) $152.05
a) $2386 b) Let $f$ = food costs; $p$ = personal and household expenses; $e[latex] = entertainment; [latex]g$ = gas. Total monthly expenses = $f+p+e+g+2386$ .
356°F
a) Total cost = $140d$ b) $280

In this section, we will take what we have learned and apply the concepts to new situations.

Perspectives

The cost of household electricity can be found by evaluating the expression $[(C - P) · E + B] ·(1+T)$ , where $C =$ current meter reading; $P =$ previous meter reading; $B =$ basic charge; $E =$ energy charge per kWh; and $T =$ tax rate.

a) What does $(C-P)$ represent in the expression?

b) What does $(C-P)\cdotE$ represent in the expression?

c) Calculate the total charge on this bill:

The cost to rent a Bouncy House for a children’s birthday party is $150 flat fee plus $30 per hour. a) If $h$ = number of rental hours, write an expression that shows the total cost of renting the bouncy house. b) Use the expression to determine the total cost of renting the bouncy house for 5 hours.
Male crickets and katydids chirp by rubbing their front wings together. Each species has its own chirp and chirping is temperature dependent. To calculate the temperature if you hear a Field Cricket, we can use temperature = 50+[(N-40)/4]. For a Snowy Tree Cricket, temperature = 50 + [(N – 92) / 4.7], and for a Katydid temperature = 60 + [(N – 19) / 3)], where N is the number of chirps per minute. a) Ben and Shana are out camping and hear crickets but have no idea what kind of crickets they are. If they count 44 chirps in 30 seconds what are the possible temperature options? b) SInce crickets generally do not sing at temperatures below 55° F or above 100 °F, what is the most probable type of cricket they are hearing? c) The simplest method is to count the number of chirps in 15 seconds and add 40. Define a variable and write an expression that shows this sum. Use the expression to determine the temperature. How close is this quick method’s temperature to your answers from part a)?
The distance an object falls under gravity can be calculated using the formula $d=\frac{1}{2}gt^2$ , where $d=$ distance in meters, $g=$ the acceleration due to gravity = 9.9 meters per square second, and $t=$ time in seconds. Calculate the depth of a canyon if a stone dropped from the top of the canyon takes 4.2 seconds to hit the floor of the canyon.
Blood alcohol content depends on many factors. To calculate the blood alcohol content of an individual we can use the formula $B=\dfrac{0.63674nda}{bF}-0.012h$ , where $B=$ blood alcohol content in ml per dl; $n=$ number of drink units, $d=$ drink volume in milliliters, $a=$ alcohol percentage by volume, $b=$ body weight in kilograms, $F=$ 0.58 for males and 0.49 for females, and $h=$ drinking time in hours. a) A 78 kg male drink 2 cans of beer. Each can measures 375 ml with a 4.55% alcohol percentage by volume. He drinks both cans within 30 minutes. Calculate his blood alcohol content. b) To convert from blood alcohol content to blood alcohol percentage, we divide the blood alcohol content by 1000. Calculate this male’s blood alcohol percentage. c) In Utah anyone driving with a 0.05% or greater blood alcohol percentage can be charged with a DUI. Is this male above or below the legal limit to drive? d) A 59 kg female drinks 2 cocktails containing 30 ml each of vodka, which has a 40% alcohol percentage by volume. She drank the cocktails over an hour and a half. Calculate her blood alcohol content. e) Convert this to blood alcohol percentage. f) Would this women be over the legal limit for driving in Utah?

Show/Hide Answer

a) The amount of electricity used. b) The cost of electricity used. c) $34.28
a) Total cost = $150+30h$ b) $300
a) Field Cricket: 62°F Snowy Tree Cricket: 49°F Katydid: 85°F b) Field Cricket or Katydid b) Let $n$ = number of chirps in 15 seconds. Temperature = $n+40$ = 66°F. This closest to the Field Cricket calculation.
$d=\frac{1}{2}(9.8)(4.2)^2=86.436$ The canyon is about 86 meters deep.
a) $B=\dfrac{0.63674(2)(375)(0.0455)}{78(0.58)}-0.012(0.5)=0.5511475$ His blood alcohol content is approximately 0.55 b) 0.055% c) above d) $B=\dfrac{0.63674(2)(60)(0.40)}{59(0.49}-0.012(1.5)=1.03919...$ Her blood alcohol content is about 1.04 e) 0.014% f) no

Skills Icon In this section, we will use what we have learned so far to practice skill problems.

Skill Exercises

Evaluate the expression:

$3a+2b-4c$ when $a=-5,\;b=4\;c= 2$
$-x(y-z)$ when $x=-4,\;y=7,\;z=-2$
$4m+7n$ when $m=2.4,\;n=-4.5$
$\dfrac{4x-y}{x+2}$ when $x=2,\;y=-4$
$\dfrac{9(C)}{5}+32$ when $C=20.5$

SImplify the expression:

$2x-7x+4x$
$-3x-5x+6y$
$4\cdot8x-3$
$3\cdot(5x-2)$
$5x-2(7x+2y)+8y$

Show/Hide Answer

–15
36
21.9
64
68.9
$-x$
$-8x+6y$
$32x-3$
$15x-6$
$-9x+4y$

Klein, F. (1924). Elementary Mathematics from an Advanced Standpoint (Translated from the 3rd German edition by Hedrick, E.R.and Noble, C.A.). Dover Publications, New York. ↵
Tarski, A. (1941). Introduction to Logic and to the Methodology of Deductive Sciences. Oxford University Press, New York) ↵

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