Elementary Algebra Expressions and Equations

Section 2.2 Expressions and Equations

Subsection Writing Algebraic Expressions

When we write an algebraic expression, we use the same operations on a variable that we would use to calculate with a specific number. Writing down the expression for a specific numerical value can help us write an algebraic expression.

Example 2.12.

Alida keeps $100 in cash from her weekly paycheck, and deposits 40% of the remainder in her savings account. If Alida’s paycheck is $p\text{,}$ write an expression for the amount she deposits in savings.

Solution.

How would we calculate Alida’s deposit if we knew her paycheck? Suppose Alida’s paycheck is $500. First she subtracts $100 from that amount to get $500-100\text{,}$ and then she takes 40% of the remainder for savings:

\begin{equation*} 0.40(\alert{500}-100)~~~~~\blert{\text{Perform operations inside parentheses first.}} \end{equation*}

If her paycheck is $p$ dollars, we perform the same operations on $p$ instead of on 500. The expression is thus

\begin{equation*} 0.40(\alert{p}-100) \end{equation*}

Reading Questions Reading Questions

1.

We write an expression with variables using the same operations we would use to calculate with a .

Answer.

specific number

2.

In Example 2.12, how does Alida calculate the amount to deposit in savings? How do we write that in an algebraic expression?

Answer.

Subtract 100 from her paycheck, then multiply the result by 0.40. $0.40(p-100)$

Subsection Evaluating Algebraic Expressions

When we evaluate an algebraic expression, we must follow the order of operations.

Example 2.13.

Four students bought concert tickets with their $20 student-discount coupons.

If the regular price of a ticket is $t$ dollars, write an expression for the total amount the four students paid.
How much did the students pay if the regular price of a ticket is $38?

Solution.

The discount price of one ticket is $t-20$ dollars. So four tickets cost

\begin{equation*} 4(t-20)~~\text{dollars} \end{equation*}
We evaluate the expression in part (a) for $t=\alert{38}\text{.}$

\begin{align*} 4(t-20) \amp = 4(\alert{38}-20) \amp \blert{\text{Simplify inside parentheses first.}}\\ \amp = 4(18)=72 \end{align*}

The students paid $72 for the tickets.

Reading Questions Reading Questions

3.

Explain why we need parentheses in Example 2.13.

Answer.

We need to subtract before we multiply.

4.

How should we write "3 times the sum of $x$ and 12"?

Answer.

$3(x+12)$

Subsection Negative Numbers

The order of operations applies to signed numbers. Operations inside parentheses or other grouping devices should be performed first.

Example 2.14.

Simplify $~~4-3-[-6+(-5)-(-2)]$

Solution.

Perform the operations inside brackets first. Simplify each step by rewriting subtractions as equivalent additions.

\begin{align*} \amp 4-3- [-6+\blert{(-5)-(-2)}] \amp\amp \blert{-5-(-2)=-5+2}\\ =\,\amp 4-3-[\blert{-6-5+2}] \amp\amp \blert{-6-5=-11;~-11+2=-9}\\ =\,\amp 4 \blert{-3-[-9]} \amp\amp \blert{\text{Rewrite as an addition.}}\\ =\,\amp 4-3+9 = 10 \amp\amp \blert{\text{Add from left to right.}} \end{align*}

Reading Questions Reading Questions

5.

Which operation is performed first in the expression $6-4x\text{?}$

Answer.

multiplication

6.

What is the first step in evaluating the expression $2(18-x)\text{?}$

Answer.

Subtract $x$ from 18

Caution 2.15.

When using negative numbers, we must be careful to distinguish between products and sums. In Example 2.16, note how the parentheses and minus signs are used in each expression.

Example 2.16.

Simplify each expression.

$\displaystyle 3(-8)$
$\displaystyle 3-(-8)$
$\displaystyle 3-8$
$\displaystyle -3-8$

Solution.

This expression is a product: $~3(-8)=-24$
This is a subtraction. We follow the rule for subtraction by changing the sign of the second number and then adding: $~3-(-8)=3+8=11$
This is an addition; the negative sign in front of tells us that we are adding $-8$ to $3\text{.}$ Thus, $~3-8=3+(-8)=-5$
This is also an addition: $~-3-8=-3+(-8)=-11$

Reading Questions Reading Questions

7.

What is wrong with this calculation: $4-3+9=4-12=-8\text{?}$

Answer.

We should subtract $4-3$ first.

Look Closer.

When we evaluate an algebraic expression at a negative number, we enclose the negative numbers in parentheses. This will help prevent us from confusing multiplication with subtraction.

Example 2.17.

Evaluate $2x-3xy$ for $x=-5$ and $y=-2\text{.}$

Solution.

We substitute $\alert{-5}$ for $x$ and $\blert{-2}$ for $y\text{,}$ then follow the order of operations.

\begin{align*} 2x-3xy \amp = 2(\alert{-5})-3(\alert{-5})(\blert{-2}) \amp \blert{\text{ Do multiplications first.}}\\ \amp = -10-(-15)(-2)\\ \amp = -10-30=-40 \end{align*}

Reading Questions Reading Questions

8.

When we evaluate an expression at a negative number, we should the negative number in .

Answer.

enclose; parentheses

Subsection Skills Warm-Up

Exercises Exercises

Exercise Group.

Choose the correct algebraic equation for each of the following situations.

\begin{align*} \amp n+12 \amp \amp 12n \amp \amp \dfrac{n}{12}\\ \amp \dfrac{12}{n} \amp \amp 12-n \amp \amp n-12 \end{align*}

1.

Helen bought $n$ packages of tulip bulbs. If each package contains 12 bulbs, how many bulbs did she buy?

2.

Henry bought a package of $n$ gladiolus bulbs, then bought 12 loose bulbs. How many bulbs did he buy?

3.

Together Karen and Dave sold 12 tickets to the spring concert. If Karen sold $n$ tickets, how many did Dave sell?

4.

Together Karl and Diana collected $n$ used books for the book sale. If Karl collected 12 books, how many did Diana collect?

5.

Greta made $n$ dollars last week. If she worked for 12 hours, how much did she make per hour?

6.

Gert jogged for 12 minutes. If she jogged $n$ miles, how many minutes does it take her to jog 1 mile?

Subsubsection Answers to Skills Warm-Up

$\displaystyle 12n$
$\displaystyle n+12$
$\displaystyle 12-n$
$\displaystyle n-12$
$\displaystyle \dfrac{n}{12}$
$\displaystyle \dfrac{12}{n}$

Subsection Lesson

Subsubsection Activity 1: Order of Operations

Exercises Exercises

Exercise Group.

Follow the order of operations to simplify each expression.

1.

$\displaystyle 5-(+7)-3-(-2)$
$\displaystyle -4-(-9)-3-8$
$\displaystyle 7(-3)-2(-5)$
$\displaystyle 9-4(-6)$

2.

$\displaystyle -6(-2)(-5)$
$\displaystyle -6(-2)-5$
$\displaystyle -6(-2-5)$
$\displaystyle -6-(2-5)$

3.

$\displaystyle [-5-8]-[7-10-4]$
$\displaystyle -6+[(4-8)-(-9)]$
$\displaystyle 28-3(-12-2 \cdot 4)$
$\displaystyle 12-36 \div 4(9-2 \cdot 3)$

Subsubsection Activity 2: Writing Algebraic Expressions

Exercises Exercises

1.

Neda decides to order some photo albums as gifts. Each album costs $12, and the total shipping cost is $4. Neda would like an algebraic expression that describes the total cost of ordering $a$ albums.

$\blert{\text{Consider some specific values for the variable:}}$

What is Neda’s bill if she orders 3 albums? If she orders 5 albums?

For $\alert{3}$ albums, the bill is:

\begin{equation*} 12 \cdot \alert{3} + 4 = \end{equation*}

For $\alert{5}$ albums, the bill is:

\begin{equation*} \hphantom{0000} \end{equation*}
Describe in words how you calculated your answers for specific values.
$\blert{\text{Replace the specific values in your calculations by a variable.}}$

If Neda orders $\alert{a}$ albums, an expression for the bill is:

Let $B$ stand for Neda’s bill, and write an equation that gives Neda’s bill, $B\text{,}$ in terms of the number of albums she orders, $a\text{:}$

2.

Megan would like to buy a kayak on sale. She calculates that the kayak she wants costs $40 less than three weeks’ salary.

Write an expression for the price of the kayak if Megan makes $280 per week.
If Megan makes $w$ dollars per week, write an expression for the price of the kayak.

3.

Emily bought five rose bushes for her garden. Each rose bush cost $9 plus tax.

Write an expression for the total amount Emily paid if the tax on one rose bush is $0.45.
If the tax on one rose bush is $t\text{,}$ write an expression for the total amount Emily paid.

Subsubsection Activity 3: Evaluating Algebraic Expressions

Exercises Exercises

1.

When a company purchases a piece of equipment such as a computer or a copy machine, the value of the equipment depreciates over time. One way to calculate the value of the equipment uses the formula

\begin{equation*} V=C\left(1-\dfrac{t}{n}\right) \end{equation*}

where $C$ is the original cost of the equipment, $t$ is the number of years since it was purchased, and $n$ stands for the useful lifetime of the equipment in years. Find the value of a 4-year-old copy machine if it has a useful lifetime of 6 years and cost $3000 when new.

List the values of the variables:
\begin{equation*} C=\fillinmath{XXXXXX},~~~t=\fillinmath{XXXXXX},~~~n=\fillinmath{XXXXXX} \end{equation*}
Substitute the values into the formula.
Simplify the expression. Follow the order of operations.

$\blert{\text{Simplify inside the parentheses.}}$

$\blert{\text{Multiply.}}$

Answer:

2.

Evaluate $~~8(x+xy)~~$ for $~x=\dfrac{1}{2}~$ and $~y=-6\text{.}$

$\blert{\text{Substitute the values:}}~~~8\left(\dfrac{1}{2}+\dfrac{1}{2}(-6)\right)$

$\blert{\text{Follow the order of operations.}}$

3.

Evaluate $~~3(a-b)~~$ for $~a=-4~$ and $~b=-6\text{.}$

4.

Evaluate $~~-2(a+b)-ab~~$ for $~a=6~$ and $~b=-3\text{.}$

Subsubsection Wrap-Up

Objectives.

In this Lesson we practiced the following skills:

Simplifying expressions with the order of operations
Writing algebraic expressions with two or more operations
Evaluating algebraic expressions at signed numbers

Questions.

Explain what is wrong with this calculation:

\begin{equation*} 9-4(-6) = 5 (-6) = -30 \end{equation*}
For "5 dollars less than the price of 3 shirts," Delbert writes $5-3s\text{.}$ What is wrong with his expression?
In problems 2 and 3 of Activity 2, which expression required parentheses? Why?
When we evaluate $ab$ for $a=6,~b=-3\text{,}$ why should we enclose $-3$ in parentheses?

Subsection Homework Preview

Exercises Exercises

Exercise Group.

Write algebraic expressions.

1.

Boyer’s history book is 600 pages long, and he reads 20 pages per night. How many pages does he have left to read after $t$ nights?

2.

Kristi deposits $50 from her paycheck into savings, and then gives herself 15% of the rest for spending money. If her paycheck is $p$ dollars, how much spending money does she get?

3.

The area of a pyramid is one-third the product of its length, its width, and its height.

4.

The difference of a number $B$ and twice its reciprocal.

Exercise Group.

Evaluate.

5.

$2m(m+p)(m-p)~~~~$ for $~m=-5~$and $~p=-8$

6.

$(z+2)(2z-1)~~~~$ for $~z=\dfrac{-3}{4}$

Subsubsection Answers to Homework Preview

$\displaystyle 600-20t$
$\displaystyle 0.15(p-50)$
$\displaystyle \dfrac{1}{3}lwh$
$\displaystyle B-\dfrac{2}{B}$
$\displaystyle 390$
$\displaystyle \dfrac{-25}{8}$

Exercises Homework 2.2

1.

The perimeter of a rectangle of length $l$ and width $w$ is given by

\begin{equation*} P=2l+2w \end{equation*}

Find the perimeter of a rectangular meeting hall with dimensions 8.5 meters by 6.4 meters.

Answer.

29.8 meters

2.

The area of a trapezoid with bases $B$ and $b$ and height $h$ is given by

\begin{equation*} A=\dfrac{1}{2}(B+b)h \end{equation*}

Find the area of a trapezoid whose bases are 9 centimeters and 7 centimeters and whose height is 3 centimeters.

3.

If a company sells $n$ items at a cost of $c$ dollars each and sells them at a price of $p$ dollars each, the company’s profit is given by

\begin{equation*} P=n(p-c) \end{equation*}

Find the profit earned by a manufacturer of bicycle equipment by selling 300 bicycle helmets that cost $32 each to produce and sell for $50 apiece.

Answer.

$5400

4.

The temperature in degrees Celsius ($\degree$C) is given by

\begin{equation*} C=\dfrac{5}{9}(F-32)\text{,} \end{equation*}

where $F$ stands for the temperature in degrees Fahrenheit ($\degree$F). Find normal body temperature in degrees Celsius if normal temperature is $98.6 \degree$ Fahrenheit.

Exercise Group.

For Problems 5–8, simplify by following the order of operations.

5.

$-18-[8-12-(-4)]$

Answer.

$-18$

6.

$3-(-6+2)+(-1-4)$

7.

$-7+[-8-(-2)]-[6+(-4)]$

Answer.

$-15$

8.

$0-[5-(-1)]+[-6-3]$

9.

Find each product.

$\displaystyle (-2)(3)(4)$
$\displaystyle (-2)(-3)(4)$
$\displaystyle (-2)(-3)(-4)$
$\displaystyle (-2)(-3)(4)(2)$
$\displaystyle (-2)(-3)(-4)(2)$
$\displaystyle (-2)(-3)(-4)(-2)$

Answer.

$\displaystyle -24$
$\displaystyle 24$
$\displaystyle -24$
$\displaystyle 48$
$\displaystyle -48$
$\displaystyle 48$

10.

Use your results from Problem 9 to complete the statements:

The product of an odd number of negative numbers is .
The product of an even number of negative numbers is .

Exercise Group.

For Problems 11–14, use the order of operations to simplify the expression.

11.

$\displaystyle -2(-3)-4$
$\displaystyle 5(-4)-3(-6)$

Answer.

$\displaystyle 2$
$\displaystyle -2$

12.

$\displaystyle (-4-3)(-4+3)$
$\displaystyle -3(8)-6(-2)-5(2)$

13.

$\displaystyle \dfrac{15}{-3}-\dfrac{4-8}{8-12}$
$\displaystyle \dfrac{4-2(-5)}{-4+3(-1)}$

Answer.

$\displaystyle -6$
$\displaystyle -2$

14.

$\displaystyle 12.6-0.32(0.25)(4.2-8.7)$
$\displaystyle (5.8-2.6)(-2.5)(-0.6+3)$

15.

Simplify each expression.

$\displaystyle -3(-4)(-5)$
$\displaystyle -3(-4))-5$
$\displaystyle -3(-4-5)$
$\displaystyle -3-(-4-5)$
$\displaystyle -3-(-4)(-5)$
$\displaystyle (-3-4)(-5)$

Answer.

$\displaystyle -60$
$\displaystyle 7$
$\displaystyle 27$
$\displaystyle 6$
$\displaystyle -23$
$\displaystyle 35$

16.

Simplify each expression.

$\displaystyle 24 \div 6-2$
$\displaystyle 24 \div (-6-2)$
$\displaystyle 24 \div (-6)-2$
$\displaystyle 24-6 \div (-2)$
$\displaystyle 24 \div (-6) \div (-2)$
$\displaystyle 24 \div (-6 \div 2)$

Answer.

Exercise Group.

For Problems 17–22, evaluate for the given values.

17.

$15-x-y~~~~$ for $x=-6,~y=8$

Answer.

$13$

18.

$p-(4-m)~~~~$ for $p=-2,~m=-6$

19.

$12x-3xy~~~~$ for $x=-3,~y=2$

Answer.

$-18$

20.

$\dfrac{y-3}{x-4}~~~~$ for $x=-9,~y=2$

21.

$\dfrac{1}{2}t(t-1)~~~~$ for $t=\dfrac{2}{3}$

Answer.

$-\dfrac{1}{9}$

22.

$\dfrac{x-m}{s}~~~~$ for $x=4,~m=\dfrac{9}{4},~s=\dfrac{3}{2}$

23.

Evaluate each expression for $x=-5\text{.}$ What do you notice?
1. $\displaystyle \dfrac{-3}{4}x$
2. $\displaystyle \dfrac{-3x}{4}$
3. $\displaystyle -0.75x$
Does $\dfrac{-8}{5}x=\dfrac{-8}{5x}\text{?}$ Support your answer with examples.

Answer.

1. $\displaystyle -\dfrac{15}{4}$
2. $\displaystyle -\dfrac{15}{4}$
3. $\displaystyle -3.75$
No

24.

Which of the following are equivalent to $-\dfrac{4}{9}x\text{?}$

$\displaystyle \dfrac{4}{9}(-x)$
$\displaystyle \dfrac{-4x}{-9}$
$\displaystyle \dfrac{-4}{9x}$
$\displaystyle \dfrac{-4x}{9}$
$\displaystyle \dfrac{-4}{9}(-x)$

Exercise Group.

For Problems 25–26, write an algebraic expression for area or perimeter.

25.

$rectngle$

Answer.

$2x+10$

26.

$rectangles$

27.

Salewa saved $5000 to live on while going to school full time. She spends $200 per week on living expenses.

How much of Salewa’s savings will be left after 3 weeks?
Describe in words how you calculated your answer to part (a).
Fill in the table below.

Number of weeks $2$ $4$ $5$

Calculation $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$

Savings left $\hphantom{00}$ $\hphantom{00}$ $\hphantom{00}$
Write an equation that gives Salewa’s savings, $S\text{,}$ after $w$ weeks.

Answer.

$4400;~ $2600
Multiply the number of weeks by $200, then subtract from $5000.
Number of weeks $2$ $4$ $5$

Savings left $4600$ $4200$ $4000$
$\displaystyle S=5000-200w$

28.

To calculate how much state income tax she owes, Francine subtracts $2000 from her income, and then takes 12% of the result.

What is Francine’s state income tax if her income is $8000?
Describe in words how you calculated your answer to part (a).
Fill in the table below.

Income $5000 $7000 $12,000

Calculation $\hphantom{0000}$ $\hphantom{0000}$ $\hphantom{0000}$

State tax $\hphantom{00}$ $\hphantom{00}$ $\hphantom{00}$
Write an equation that gives Francine’s state income tax $T\text{,}$ if her income is $I\text{.}$

Exercise Group.

For Problems 29–32,

Choose a variable for the unknown quantity and write an algebraic expression.
Evaluate the expression for the given values.

29.

Three inches less than twice the width
The width is 13 inches.

Answer.

$w=~$ width (inches);$~~2w-3$
23 inches

30.

Twenty dollars more than 40% of the principal
The principal is $500.

31.

$8 times the number of children’s tickets subtracted from 150
There are 83 children’s tickets

Answer.

$t=~$ number of children’s tickets; $8(150-t)$
$536

32.

One-third of $50 less than the profit
The profit is $500

Exercise Group.

For Problems 33–40, write an algebraic expression.

33.

The oven temperature started at $65 \degree$ and is rising at $30 \degree$ per minute. Write an expression for the oven temperature after $t$ minutes.

Answer.

$65+30t$

34.

Luisa’s parents have agreed to pay her tuition ($800 per year) plus half her annual living expenses while she is in school. Write an expression for the amount her parents will pay if Luisa’s annual expenses are $a$ dollars.

35.

Mildred canned 80 pints of tomatoes. She kept some for herslf and divided the rest equally among her four daughters. If Mildred kept $M$ pints, write an expression for the number of pints she gave each daughter.

Answer.

$\dfrac{80-M}{4}$

36.

Moira’s income is $50 more than one-third of her mother’s income. Write an expression for Moira’s income if her mother’s income is $I\text{.}$

37.

Otis buys 200 pounds of dog food and uses 15 pounds a week for his dog Ralph. Write an expression for the amount of dog food Otis has left after $w$ weeks.

Answer.

$200-15w$

38.

Renee receives $600 for appearing in a cola commercial, plus a residual of $80 each time the commercial is aired. Write an expression for Renee’s earnings if the commercial plays $t$ times.

39.

Digby bought scuba diving gear by making a $50 down payment and arranging to pay the balance in ten equal installments. If the total cost of the gear is $C$ dollars, write an expression for the amount of each installment.

Answer.

$\dfrac{c-50}{10}$

40.

Each passengers and three crew members on a small airplane is allowed 35 pounds of luggage. Write an expression for the weight of the luggage if there are $x$ passengers.

Exercise Group.

For Problems 41–42,

Write an algebraic expression.
Make a table for the expression showing at least two positive values for the variable and two negative values.

41.

The ratio of a number to 4 more than the number

Answer.

$\displaystyle \dfrac{x}{4+x}$
Many answers possible

42.

The sum of $a$ and 5, times the difference of $a$ and 5

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Number of weeks	\(2\)	\(4\)	\(5\)
Calculation	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
Savings left	\(\hphantom{00}\)	\(\hphantom{00}\)	\(\hphantom{00}\)

Number of weeks	\(2\)	\(4\)	\(5\)
Savings left	\(4600\)	\(4200\)	\(4000\)

Income	$5000	$7000	$12,000
Calculation	\(\hphantom{0000}\)	\(\hphantom{0000}\)	\(\hphantom{0000}\)
State tax	\(\hphantom{00}\)	\(\hphantom{00}\)	\(\hphantom{00}\)