Skip to main content
Logo image

Open Resources for Community College Algebra

Faculty

🔗

Section 1.1 Variables and Evaluating Expressions

🔗
Variables and expressions are the basic building blocks for writing algebra. In this section, we explore how to use them.
🔗
Figure 1.1.1. Alternative Video Lesson
🔗

Subsection 1.1.1 Introduction to Variables

🔗
When we want to represent an unknown quantity or a quantity whose value can change, we use a variable. For example, if you’d like to write about automobile gas mileage, you could use the symbol “g” as a variable to represent a car’s gas mileage. The gas mileage g might be 25 mpg (miles per gallon) for one car, 30 mpg for some other car, or other values for other cars. It might be one thing for your car when it was new, and something else ten years later.
🔗
Since we are using a variable, we can discuss gas mileage for Honda Civics, Ford Explorers, and all other makes and models at the same time, even though these makes and models each have their own gas mileage.
🔗
When variables stand for physical quantities, it’s good to use letters that clearly represent those quantities. For example, it is wise to use g for gas mileage. This helps people who read your mathematical writing understand it better. It is common to use x, y, and z for variables when there is no context to suggest something more meaningful like g. You may see the variable x a lot.
🔗
It is important to be clear about what unit of measure goes with a variable. With gas mileage g, if we all agree to use mpg for its units, then g might be a placeholder for 25, 30, etc. On the other hand if we decide to use kpg (kilometers per gallon) for units, those quantities would be 40, 48, etc. So it’s important to tell readers that g represents gas mileage in miles per gallon or kilometers per gallon or whatever the case may be.
🔗
Sometimes the units we should use for a variable are suggested indirectly. For example if we’re told that a car has used so many gallons of gas after traveling so many miles, then we should measure gas mileage in mpg, not kpg.
🔗

Checkpoint 1.1.2. Naming Variables.

🔗
Identify a variable you might use to represent each quantity. Then identify what units would be most appropriate.
🔗
(a)
🔗
Remi is studying college student demographics, including their ages. Let be the age of a student, measured in .
Explanation.
The unknown quantity is age, which we generally measure in years. So we could say “Let a be the age of a student, measured in years.”
🔗
(b)
🔗
Luka needs to drive from Portland, OR to Boise, ID. Let be the amount of time passed since Luka left Portland, measured in .
Explanation.
The amount of time passed is the unknown quantity. Since this is a drive from Portland to Boise, it makes sense to measure this in hours, not minutes or weeks. So we could say “Let t be the amount of time passed since Luka left Portland, measured in hours.”
🔗
(c)
🔗
There is a number that you will triple, then add five to, and then take the square root. Let be this number.
Explanation.
The number we are discussing doesn’t have any physical context, so we choose to use the generic variable x to represent it.
🔗

Subsection 1.1.2 Algebraic Expressions

🔗
🔗
Any combination of variables and numbers using arithmetic operations (like addition, multiplication, etc.) is an algebraic expression. The following are examples of algebraic expressions:
x+12ℓ+2wxy+1nRT
🔗
Note that this definition of “algebraic expression” does not include anything with an equal sign (=) in it. The idea of an equation (which has an equals sign) is discussed in Section 4.
🔗

Example 1.1.3.

🔗
🔗
The expression:
59(F−32)
🔗
converts a temperature in degrees Fahrenheit to degrees Celsius. To do this, we need a Fahrenheit temperature, F. Then we can evaluate the expression. This means replacing its variable(s) with specific numbers and finding the result as a single, simplified number.
🔗
🔗
Let’s convert the temperature 89℉ to the Celsius scale by evaluating the expression. To do this, we substitute the number 89 in place of the variable F.
59(89−32)=59(57)Review order of operations in [cross−referencetotarget(s)"section−order−of−operations"missingornotunique].=2859Review fraction multiplication in [cross−referencetotarget(s)"section−fractions−and−fraction−arithmetic"missingornotunique].≈31.67
🔗
This shows us that 89℉ is approximately the same as 31.67℃.
🔗

Warning 1.1.4. Vocabulary.

🔗
The steps in Example 3 are not “solving”, as far as algebra vocabulary is concerned. “Solving” is a word you might want to use because in everyday English you are “finding an answer”. However in algebra, there is a special meaning for the term “solving” that is discussed in Section 5. Here, when we substitute in values for variables and then compute the result, we are “evaluating an expression”, not solving anything.
🔗

Checkpoint 1.1.5. Convert Temperature.

🔗
Try evaluating the temperature expression for yourself.
🔗
(a)
🔗
If a temperature is 50℉, what is that temperature in Celsius?
Explanation.
59(50−32)=59(18)=51(2)=10
So 50℉ is equivalent to 10℃.
🔗
(b)
🔗
If a temperature is −20℉, what is that temperature in Celsius?
Explanation.
59(−20−32)=59(−52)=−2609≈−28.89
So −20℉ is equivalent to about −28.89℃.
🔗

Example 1.1.6. Stair Rise and Run.

🔗
When building a staircase, you want the rise and run to be consistent from one step to the next.
diagram of a staircase with one stair’s height labeled as ’rise’ and its width labeled as ’run’
🔗
A convention among contractors is that a staircase run, in inches, is given by 17.5−h where h is the rise in inches.
🔗
(a)
🔗
Determine the run for each step of a staircase where the rise is 7 in.
Explanation.
We substitute 7 for h:
17.5−7=10.5.
So the run is 10.5 in.
🔗
(b)
🔗
A staircase needs to span a total height of 108 in. What is a reasonable number of steps for it to have? What will that mean for the rise of each step? What will the run be for each step?
Explanation.
There is more than one good answer, but if there are 12 steps then the height of each step will be 10812 inches, or 9 in. And that mean the run of each step is found by substituting 9 for h:
17.5−9=8.5.
So each step would have a run of 8.5 in.
🔗

Checkpoint 1.1.7. Stair Rise and Run.

🔗
The expression 17.5−h gives the run of a stairstep when the rise is h inches. Determine the run for each step of a staircase where the rise is 7.75 in.
Explanation.
We substitute 7.75 for h:
17.5−7.75=9.75.
So the run is 9.75 in.
🔗

Checkpoint 1.1.8. Rising Rents.

🔗
In Oregon starting from the year 2000, the median rent among all rental living units has closely followed the expression 620+32.35x, where x is the number of years since the year 2000.
🔗
(a)
🔗
According to this model, what was the median rent for a living unit in Oregon in 2010?
Explanation.
This model uses x as the number of years since 2000. So for the year 2010, x is 10:
620+32.35(10)=943.50
According to this model, the median monthly rent for a living unit in Oregon in 2010 was $943.50.
🔗
(b)
🔗
According to this model, what was the median rent for a living unit in Oregon in 2020?
Explanation.
For the year 2020, x is 20:
620+32.35(20)=1267
According to this model, the median monthly rent for a living unit in Oregon in 2020 was $1267.
🔗
(c)
🔗
According to this model, what will be the median rent for a living unit in Oregon in 2030?
Explanation.
For the year 2030, x is 30:
620+32.35(30)=1590.50
According to this model, the median monthly rent for a living unit in Oregon in 2030 will be $1590.50.
🔗

Subsection 1.1.3 Evaluating Expressions with Exponents, Absolute Value, and Radicals

🔗
Algebraic expressions might have exponents, absolute value bars, and radicals. This does not change the basic approach to evaluating them.
🔗

Example 1.1.9. Tsunami Speed.

🔗
The speed of a tsunami (in meters per second) can be modeled by 9.8d, where d is the depth of the tsunami (in meters). Determine the speed of a tsunami that has a depth of 30 m to four significant digits.
Explanation.
Using d=30, we find:
9.8(30)=294Review order of operations in [cross−referencetotarget(s)"section−order−of−operations"missingornotunique].≈17.14⏞four6428…Review square root in [cross−referencetotarget(s)"section−absolute−value−and−square−root"missingornotunique].
The speed of tsunami with a depth of 30 m is about 17.15 m⁄s.
🔗
We have been evaluating expressions, but we can evaluate formulas in the same way. A formula has an equal sign with an expression to the right. On the left of the equal sign, there is a variable that represents the result. For example, we could write the formula s=9.8d for the speed of a tsunami from Example 9.
🔗

Checkpoint 1.1.10. Tent Height.

🔗
The height inside a tent when you are d feet from the west wall of the tent is given by the formula h=−2|d−3|+6, where h is in feet.
an A-frame tent with guidelines indicating that d feet inward from one side, the height to the ceiling at that point is d feet
🔗
(a)
🔗
When you are 5 ft from the west side, the height is .
Explanation.
When d=5, we have:
h=−2|d−3|+6h=−2|5−3|+6Review order of operations in [|[cross−referencetotarget(s)"section−order−of−operations"missingornotunique]|]∗.=−2|2|+6Review absolute value in [|[cross−referencetotarget(s)"section−absolute−value−and−square−root"missingornotunique]|]∗.=−2(2)+6=−4+6=2
So when you are 5 ft from the west side, the height of the tent is 2 ft.
🔗
(b)
🔗
When you are 2.5 ft from the west side, the height is .
Explanation.
When d=2.5, we have:
h=−2|d−3|+6h=−2|2.5−3|+6=−2|−0.5|+6=−2(0.5)+6=−1+6=5
So when you are 2.5 ft from the west side, the height of the tent is 5 ft.
🔗

Checkpoint 1.1.11. Mortgage Payments.

🔗
If we borrow L dollars for a home mortgage loan at an annual interest rate r, and intend to pay off the loan after n months, then the amount we should pay each month M, in dollars, is given by the formula
🔗
M=rL(1+r12)n12((1+r12)n−1)
🔗
If we borrow $200,000 at an interest rate of 6% with the intent to pay off the loan in 30 years, what should our monthly payment be? (Using a calculator is appropriate here.)
Explanation.
We must use L=200000. The interest rate r is a percentage, so we write r=0.06 (not r=6). The variable n is supposed to be a number of months, but we will pay off the loan in 30 years. Therefore we take n=360.
M=(0.06)(200000)(1+0.0612)36012((1+0.0612)360−1)=(0.06)(200000)(1+0.005)36012((1+0.005)360−1)≈(0.06)(200000)(6.022575…)12(6.022575…−1)≈(0.06)(200000)(6.022575…)12(5.022575…)≈72270.90…60.2709…≈1199.10
Our monthly payment should be $1,199.10.
🔗

Warning 1.1.12. Rounding Too Much.

🔗
You might have noticed in the explanation to Checkpoint 11 that during the computations, many decimal places were recorded at each step. Tracking lots of decimal places might be important, depending on what you are working toward. If you round in the middle of your work, you have changed the numbers a little bit from what they really should be. As computation continues, this little error can become larger and larger, leaving you with a final result that is too far off from correct. So the best practice is to keep lots of decimal places in all your computations, and then at the very end you may round more if that is appropriate.
🔗

Subsection 1.1.4 Evaluating Expressions with Negative Numbers

🔗
When we substitute negative numbers into an expression, it’s important to use parentheses around them or else it’s easy to forget that a negative number is being raised to a power.
🔗

Example 1.1.13.

🔗
Evaluate x2 for x=−2.
🔗
🔗
We substitute:
x2=(−2)2=4
🔗
If we don’t use parentheses, we would have:
x2=−22incorrect!=−4
🔗
The original expression x2 takes x and squares it, so we want to do the same thing to the number −2. But with the incorrect expression −22, the number −2 is not being squared. An exponent has higher priority than negation in the order of operations, so −22 is the same as −(22), and the wrong number is being squared. With (−2)2 the number −2 is being squared, which is what we want.
🔗
So it’s wise to always use parentheses when substituting in a negative number.
🔗

Checkpoint 1.1.14. Multivariable Expressions.

🔗
Evaluate the following expressions for x=−2 and y=−5. You may review multiplying negative numbers in [cross-reference to target(s) "section-arithmetic-with-negative-numbers" missing or not unique].
🔗
(a)
🔗
x3y2
Explanation.
(−2)3(−5)2=(−8)(25)=−200
🔗
(b)
🔗
(−2x)3
Explanation.
(−2(−2))3=(4)3=64
🔗
(c)
🔗
−3x2y
Explanation.
−3(−2)2(−5)=−3(4)(−5)=60
🔗

Reading Questions 1.1.5 Reading Questions

🔗

1.

🔗
What is a reason for wanting to use a letter other than x, y, or z as a variable?
🔗

2.

🔗
What is the difference between an “algebraic expression” and a “formula”, as these things were described in this section? (Other math resources may define these terms differently.)
🔗

3.

🔗
What should you watch out for when substituting in a negative number for a variable?
🔗

Exercises 1.1.6 Exercises

🔗

Skills Practice

🔗
Naming Variables.
🔗
Identify a variable you might use to represent each quantity. Then identify what units would be most appropriate.
🔗
1.
🔗
(a)
🔗
Let be the depth of a swimming pool, measured in .
🔗
(b)
🔗
Let be the weight of a dog, measured in .
🔗
2.
🔗
(a)
🔗
Let be the amount of time a person sleeps each night, measured in .
🔗
(b)
🔗
Let be the surface area of a patio, measured in .
🔗
Evaluating Expressions.
🔗
Evaluate the expression for the given value of the variable.
🔗
27.
🔗
6(x+8)+9 for x=−4
🔗
28.
🔗
−(z−1)+5 for z=−5
🔗
29.
🔗
−−7c+82c for c=−4
🔗
30.
🔗
−5f−37f for f=−4
🔗
31.
🔗
5h+7l for h=−5, l=4
🔗
32.
🔗
−6k+3b for k=−4, b=−5
🔗
33.
🔗
n2−t4 for n=−5, t=6
🔗
34.
🔗
−r9−j8 for r=−4, j=−3
🔗
35.
🔗
−18(m+1)2+5 for m=−9.
🔗
36.
🔗
−13(p−7)2 for p=4.
🔗
37.
🔗
−4q2−5q−6 for q=4.
🔗
38.
🔗
4y2+8y−6 for y=9.
🔗
39.
🔗
−7r2+3r−6 for r=−4.
🔗
40.
🔗
−2a2−6a−5 for a=−7.
🔗
41.
🔗
(9b)2 for b=−1.
🔗
42.
🔗
(−4A)3 for A=−4.
🔗
44.
🔗
(−6m)2 for m=9.
🔗
46.
🔗
q−9+4 for q=13.
🔗
47.
🔗
−(8y−1+1) for y=65.
🔗
48.
🔗
−(8r+7+6) for r=18.
🔗
49.
🔗
|a−4|+7 for a=−6.
🔗
50.
🔗
|b+4|+2 for b=6.
🔗
51.
🔗
−9|A−7|−3 for A=−1.
🔗
52.
🔗
−9|B+1|−8 for B=−7.
🔗
53.
🔗
y2−y1x2−x1 for x1=5, x2=6, y1=9, and y2=5.
🔗
54.
🔗
y2−y1x2−x1 for x1=8, x2=−2, y1=−3, and y2=1.
🔗
55.
🔗
(x2−x1)2+(y2−y1)2 for x1=6, x2=−2, y1=10, and y2=16.
🔗
56.
🔗
(x2−x1)2+(y2−y1)2 for x1=−6, x2=2, y1=9, and y2=3.
🔗

Applications

🔗
Celsius to Kelvin.
🔗
To convert a temperature in degrees Celsius to degrees Kelvin, there is a formula K=C+273.
🔗
57.
🔗
What Kelvin temperature is 7℃?
🔗
58.
🔗
What Kelvin temperature is 12℃?
🔗
Age of the United States.
🔗
To find the age of the United States in year Y, there is a formula A=Y−1776.
🔗
59.
🔗
How old was the United States in the year 1950?
🔗
60.
🔗
How old was the United States in the year 1963?
🔗
Portland to Boise.
🔗
If you travel the road from Portland, OR to Boise, ID, and you have traveled x miles so far, you have 431−x miles left to go.
🔗
61.
🔗
After traveling 227 miles, how far do you have left to go?
🔗
62.
🔗
After traveling 250 miles, how far do you have left to go?
🔗
Hours and Seconds.
🔗
There is a formula to convert a number of hours into so many seconds: s=3600h.
🔗
63.
🔗
How much time is 21 hours in seconds?
🔗
64.
🔗
How much time is 24 hours in seconds?
🔗
Celsius to Fahrenheit.
🔗
To convert a temperature in degrees Celsius to degrees Kelvin, there is a formula F=95C+32.
🔗
65.
🔗
What Fahrenheit temperature is 3℃?
🔗
66.
🔗
What Fahrenheit temperature is 7℃?
🔗
Target Heart Rate.
🔗
If we want to represent a person’s target heart rate during exercise, we’d use the formula r=0.6(220−a) where a is the person’s age in years and r is their target heart rate in beats per minute (bpm).
🔗
67.
🔗
What is the target heart rate of a person who is 35 years old?
🔗
68.
🔗
What is the target heart rate of a person who is 40 years old?
🔗
Wealth Distribution.
🔗
Since the year 2010, the percent of wealth in the United States that is held by the wealthiest 1% has followed the formula p=0.339(n−2010)+28.3 where n is the year.
🔗
69.
🔗
What percent of wealth in the United States was held by the wealthiest 1% in the year 2016?
🔗
70.
🔗
What percent of wealth in the United States was held by the wealthiest 1% in the year 2017?
🔗
Throw a Baseball.
🔗
On Earth, if you throw a baseball straight up at speed v (in feet per second), the height of the ball t seconds later is given by −16t2+vt+6.
🔗
71.
🔗
If thrown straight up with speed 79 feet per second, how high up is the baseball after 3 seconds?
🔗
72.
🔗
If thrown straight up with speed 84 feet per second, how high up is the baseball after 1 seconds?
🔗
High Point.
🔗
On Earth, if you throw a baseball straight up at speed v (in feet per second), the highest that it reaches is v2/64+6 feet above the ground.
🔗
73.
🔗
If thrown straight up with speed 89 feet per second, how high will the baseball reach?
🔗
74.
🔗
If thrown straight up with speed 48 feet per second, how high will the baseball reach?
🔗
The Baseball Lands.
🔗
On Earth, if you throw a baseball straight up at speed v (in feet per second), it will land after v+v2+38432 seconds.
🔗
75.
🔗
If thrown straight up with speed 53 feet per second, how long will it be until the baseball hits the ground?
🔗
76.
🔗
If thrown straight up with speed 58 feet per second, how long will it be until the baseball hits the ground?
🔗
Cesarean Delivery.
🔗
The percentage of births in the U.S. delivered via cesarean section in a year can be given by the formula p=0.0042x3−0.0529x2+0.0212x+32.842 where x is the number of years since 2010.
🔗
77.
🔗
What percent of births in the US in the year 2015 were cesarean deliveries?
🔗
78.
🔗
What percent of births in the US in the year 2016 were cesarean deliveries?
🔗
Diagonal of a Rectangle.
🔗
The diagonal length D of a rectangle with side lengths L and W is given by D=L2+W2.
rectangle with sides labeled L and W, and diagonal labeled D
🔗
79.
🔗
Determine the diagonal length of a rectangle with side lengths 24 and 10.
🔗
80.
🔗
Determine the diagonal length of a rectangle with side lengths 21 and 20.
🔗
Tent Height.
🔗
The height inside a tent when you are d feet from the west side of the tent is given by the formula h=−2|d−3|+6, where h is in feet.
an A-frame tent with guidelines indicating that d feet inward from one side, the height to the ceiling at that point is d feet
🔗
81.
🔗
(a)
🔗
When you are 2.2 ft from the west side, how high up is the ceiling?
🔗
(b)
🔗
When you are 2.6 ft from the west side, how high up is the ceiling?
🔗
82.
🔗
(a)
🔗
When you are 2.5 ft from the west side, how high up is the ceiling?
🔗
(b)
🔗
When you are 4.6 ft from the west side, how high up is the ceiling?
🔗
83.
🔗
There are 60 seconds in a minute. Write an expression representing how many seconds are in x minutes.
🔗
84.
🔗
There are 12 inches in a foot. Write an expression representing how many inches are in x feet.
🔗
85.
🔗
If a population of x fish is introduced into a large, predator-free lake with ample food, then one year later the population will have grown by 80 percent. Find an expression for the population one year later.
🔗
86.
🔗
If a pool with x gallons of water is left uncovered outside for one month with no rain, it will lose 11 percent of its volume to evaporation. Find an expression for the number of gallons after one month.
🔗
87.
🔗
Suppose n family members live in a home, and some cousins, a family of five, comes to stay for a week. Give an expression for how many people live in that house during that week.
🔗
88.
🔗
Back home in your refrigerator are some eggs. There are n eggs, but you cannot recall how many. While shopping, you purchase a dozen eggs. Write an expression for how many eggs you have when you get home.
🔗
89.
🔗
The rental fee for a beach house is a flat $360 plus $125 per night. Write an expression for the cost to rent the beach house for n nights.
🔗
90.
🔗
An elementary school classroom needs a minimum of 140 square feet for the teacher plus a minimum of 44 square feet per student. Write an expression for the minimum square footage of a classroom with n students.
You have attempted 1 of 101 activities on this page.
PrevTopNext