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Section 1.2 Algebraic Expressions

Subsection Writing an Algebraic Expression

An algebraic expression, or simply an expression, is any meaningful combination of numbers, variables, and operation symbols.

For example,

\begin{equation*} 4 \times g,~~~~~~3 \times c + p,~~~~~~\text{and}~~~~~~\dfrac{n-7}{w} \end{equation*}

are algebraic expressions. An important part of algebra involves translating word phrases into algebraic expressions.

Example 1.6.

Write an algebraic expression for each quantity.

  1. 30% of the money invested in stocks
  2. The cost of the dinner split three ways
Solution
  1. The amount invested is unknown, so we choose a variable to represent it.

    \begin{equation*} \blert{\text{Amount invested in stocks:}~~~s} \end{equation*}

    Next, we identify the operation described: "30% of" means 0.30 times:

    \begin{equation*} \text{The expression is} ~~~~\blert{0.30s} \end{equation*}
  2. The cost of the dinner is unknown, so we choose a variable to represent it.

    \begin{equation*} \blert{\text{Cost of dinner:}~~~C} \end{equation*}

    Next, we identify the operation described: "Split" means divided:

    \begin{equation*} \text{The expression is} ~~~~\blert{\dfrac{C}{3}} \end{equation*}

In In Example 1.6, we used three steps to write an algebraic expression.

To write an algebraic expression.
  1. Identify the unknown quantity and write a short phrase to describe it.
  2. Choose a variable to represent the unknown quantity.
  3. Use mathematical symbols to represent the relationship.

Reading Questions Reading Questions

1.

A meaningful combination of numbers, variables and operation symbols is called an .

Subsection Sums

When we add two numbers \(a\) and \(b\text{,}\) the result is called the sum of \(a\) and \(b\text{.}\) We call the numbers \(a\) and \(b\) the terms of the sum.

sums
Example 1.7.

Write sums to represent the following phrases.

  1. Six more than \(x\)

  2. Fifteen greater than \(r\)

Solution
  1. \(6+x\) or \(x+6\)

  2. \(15+r\) or \(r+15\)

The terms can be added in either order. We call this fact the commutative law for addition.

Commutative Law for Addition.

If \(a\) and \(b\) are numbers, then

\begin{equation*} \blert{a+b = b+a} \end{equation*}

Subsection Products

When we multiply two numbers \(a\) and \(b\text{,}\) the result is called the product of \(a\) and \(b\text{.}\) The numbers \(a\) and \(b\) are the factors of the product.

Look Closer.

In arithmetic we use the symbol \(\times\) to denote multiplication. However, in algebra, \(\times\) may be confused with the variable \(x\text{.}\) So, instead, we use a dot or parentheses for multiplication, like this:

products

If one of the factors is a variable, we can just write the factors side by side. For example,

\begin{equation*} \begin{aligned} ab~~~~\amp \text{means} \amp\amp \text{the product of}~~ a~~ \text{times}~~ b\\ 3x~~~~\amp \text{means} \amp\amp \text{the product of}~~ 3~~ \text{times}~~ x\\ \end{aligned} \end{equation*}
Example 1.8.

Write products to represent the following phrases.

  1. Ten times \(z\)
  2. Two-thirds of \(y\)
  3. The product of \(x\) and \(y\)
Solution
  1. \(10z\)
  2. \(\dfrac{2}{3}y\)
  3. \(xy\) or \(yx\)
Look Closer.

In Example 1.8a, \(10z = 10 \cdot z = z \cdot 10\text{,}\) because two numbers can be multiplied in either order to give the same answer. This is the commutative law for multiplication.

Commutative Law for Multiplication.

If \(a\) and \(b\) are numbers, then

\begin{equation*} \blert{a \cdot b = b \cdot a} \end{equation*}

In algebra, we usually write products with the numeral first. Thus, we write \(10z\) for "\(10\) times \(z\)”.

Reading Questions Reading Questions

2.

When we add two numbers, the result is called the and the two numbers are called the .

3.

When we multiply two numbers, the result is called the and the two numbers are called the .

Subsection Differences

When we subtract \(b\) from \(a\) the result is called the difference of \(a\) and \(b\text{.}\) As with addition, \(a\) and \(b\) are called terms.

differences
Caution 1.9.

The difference \(5-2\) is not the same as \(2-5\text{.}\) When we translate "\(a\) subtracted from \(b\text{,}\)" the order of the terms is important.

\begin{equation*} \begin{aligned} b-a~~~~\amp \text{means} \amp\amp a~~ \text{subtracted from}~~ b\\ x-7~~~~\amp \text{means} \amp\amp 7~~ \text{subtracted from}~~ x\\ \end{aligned} \end{equation*}

The operation of subtraction is not commutative.

Subsection Quotients

When we divide \(a\) by \(b\) the result is called the quotient of \(a\) and \(b\text{.}\) We call \(a\) the dividend and \(b\) the divisor. In algebra we indicate division by using the division symbol, \(\div\text{,}\) or a fraction bar.

quotients
Look Closer.

The operation of division is not commutative. The order of the numbers in a quotient makes a difference. For example, \(12 \div 3 = 4\) but \(3 \div 12 = \dfrac{1}{4}\text{.}\)

Example 1.10.

Write an algebraic expression for each phrase.

  1. \(z\) subtracted from \(13\)
  2. \(25\) divided by \(R\)
Solution
  1. \(13-z\)
  2. \(\dfrac{25}{R}\)

Reading Questions Reading Questions

4.

Which operations obey the commutative laws? What do these laws say?

5.

When we subtract two numbers, the result is called the .

6.

When we divide two numbers, the result is called the .

Subsection Evaluating an Algebraic Expression

An algebraic expression is a pattern or rule for different versions of the situation it describes. We can replace the variable by specific numbers to fit a particular situation.

Substituting a specific value into an expression and calculating the result is called evaluating the expression.

Example 1.11.

Fernando has three roommates, and his share of the rent is \(\dfrac{R}{4}\text{,}\) where \(R\) is a variable representing the total rent on the apartment. If Fernando and his roommates find an apartment whose rent is $540, we can find Fernando's share of the rent by replacing \(R\) by \(\alert{540}\text{.}\)

\begin{equation*} \dfrac{R}{4} = \alert{540} \div 4 = 135 \end{equation*}

Fernando's share of the rent is $135.

In Example 1.11, we evaluated the expression \(\dfrac{R}{4}\) for \(R=540\text{.}\)

Subsection Some Useful Algebraic Formulas

Here are some examples of common formulas. Formulas may involve several different variables.

To find the distance traveled by an object moving at a constant speed or rate for a specified time, we multiply the rate by the time.

\begin{equation*} \blert{\text{distance} = \text{rate} \times \text{time}~~~~~~~~~~~~~d = rt} \end{equation*}
Example 1.12.

How far will a train moving at 80 miles per hour travel in three hours?

Solution

We evaluate the formula with \(r=\alert{80}\) and \(t=\alert{3}\) to find

\begin{equation*} \begin{aligned} d \amp = rt\\ \amp = (\alert{80})(\alert{3}) = 240 \end{aligned} \end{equation*}

The train will travel 240 miles.

To find the profit earned by a company we subtract the costs from the revenue.

\begin{equation*} \blert{\text{profit} = \text{revenue} - \text{cost}~~~~~~~~~~~~~P = R-C} \end{equation*}
Example 1.13.

The owner of a sandwich shop spent $800 last week for labor and supplies. She received $1150 in revenue. What was her profit?

Solution

We evaluate the formula with \(R=\alert{1150}\) and \(C=\alert{800}\) to find

\begin{equation*} \begin{aligned} P \amp = R-C\\ \amp = \alert{1150}-\alert{350} = 350 \end{aligned} \end{equation*}

The owner's profit was $350.

To find the interest earned on an investment we multiply the amount invested (the principal) by the interest rate and the length of the investment.

\begin{equation*} \blert{\text{interest} = \text{principal} \times \text{interest rate} \times \text{time}~~~~~~~~~~~~~I=Prt} \end{equation*}
Example 1.14.

How much interest will be earned on $800 invested for 3 years in a savings account that pays \(5\dfrac{1}{2}\)% interest?

Solution

We evaluate the formula with \(P=\alert{800}\text{,}\) \(r=\alert{0.055}\text{,}\) and \(t=\alert{3}\) to find

\begin{equation*} \begin{aligned} I \amp = Prt\\ \amp = \alert{800} \times \alert{0.055} \times \alert{3} = 132 \end{aligned} \end{equation*}

The account will earn $132 in interest.

To find a percentage or part of a given amount, we multiply the whole amount by the percentage rate.

\begin{equation*} \blert{\text{part} = \text{percentage rate} \times \text{whole} ~~~~~~~~~~~~~P=rW} \end{equation*}
Example 1.15.

How much ginger ale do you need to make 60 gallons of a fruit punch that is 20% ginger ale?

Solution

We evaluate the formula with \(r=\alert{0.20}\) and \(W=\alert{60}\) to find

\begin{equation*} \begin{aligned} P \amp = rW\\ \amp = \alert{0.20} \times \alert{60} = 12 \end{aligned} \end{equation*}

You would need 12 gallons of ginger ale.

To find the average of a collection of scores, we divide the sum of the scores by the number of scores.

\begin{equation*} \blert{\text{average score} = \dfrac{\text{sum of scores}}{\text{number of scores}} ~~~~~~~~~~~~~A=\dfrac{S}{n}} \end{equation*}
Example 1.16.

Bert's quiz scores in chemistry are 15, 16, 18, 18, and 12. What is his average score?

Solution

We evaluate the formula with \(S=\alert{15+16+18+18+12}\) and \(n=\alert{5}\) to find

\begin{equation*} \begin{aligned} A \amp = \dfrac{S}{n}\\ \amp = \dfrac{\alert{15+16+18+18+12}} {\alert{5}} = 15.8 \end{aligned} \end{equation*}

Bert's average score is 15.8.

Reading Questions Reading Questions

7.

What does it mean to evaluate an expression?

8.

How do we change a percent to a decimal fraction?

Look Ahead.

You will also need to know the formulas for the area and perimeter of a rectangle. See the Skills Warm-Up to review these formulas.

Subsection Skills Warm-Up

Exercises Exercises

Recall the formulas for the area and perimeter of a rectangle:

\begin{equation*} \blert{A = lw ~~~~~~~~\text{and} ~~~~~~~~P=2l+2w} \end{equation*}
1.

Delbert's living room is 20 feet long and 12 feet wide. How much oak baseboard does he need to border the floor? (Don't worry about doorways.)

2.

How much wood parquet tiling must he buy to cover the floor?

3.

The distance you jog around the shore of a small lake

4.

The amount of Astroturf needed for the new football field

5.

The amount of grated cheese needed to cover a pizza

6.

The number of tulip bulbs needed to border a patio

7.
stacks of squares
8.
polygonal region with right angles

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 1.2

For Problems 1–10, write the word phrase as an algebraic expression.

1.

Product of 4 and \(y\)

2.

Twice \(b\)

3.

115% of \(g\)

4.

\(t\) decreased by 5

5.

The quotient of 7 and \(w\)

6.

4 divided into \(B\)

7.

20 more than \(T\)

8.

16 reduced by \(p\)

9.

The ratio of 15 to \(M\)

10.

The difference of \(R\) and 3.5

For Problems 11–14, choose the correct algebraic expression.

  • \(n+6\)
  • \(n-6\)
  • \(6-n\)
  • \(6n\)
  • \(\dfrac{n}{6}\)
  • \(\dfrac{6}{n}\)
11.

Rashad is 6 years younger than Shelley. If Shelley is \(n\) years old, how old is Rashad?

12.

Each package of sodas contains 6 cans. If Antoine bought \(n\) packages of sodas, how many cans did he buy?

13.

Lizette and Patrick together own 6 cats. If Lizette owns \(n\) cats, how many cats does Patrick own?

14.

Mitra divided 6 cupcakes among \(n\) children. How much of a cupcake did each child get?

For Problems 15–17, write an algebraic expression for the area or perimeter of the figure. Include units in your answers. The dimensions are given in inches.

15.

area =

16.

perimeter =

17.

area =

18.
  1. Which of these expressions says "one-sixth of \(x\)" ?

    \begin{equation*} \dfrac{x}{6}~~~~~~\dfrac{6}{x}~~~~~~\dfrac{1}{6}x~~~~~\dfrac{1}{6x} \end{equation*}
  2. Which of these expressions says "6% of \(x\)" ?

    \begin{equation*} 0.6x~~~~~~0.06x~~~~~~\dfrac{6}{100}x~~~~~~\dfrac{1}{6}x \end{equation*}
19.

Francine is saving up to buy a car, and deposits \(\dfrac{1}{5}\) of her spending money each month into a savings account.

  1. If \(m\) stands for Francine's spending money, write an expression for the amount she saves.
  2. Evaluate the expression to complete the table.
    Spending money (dollars) \(20\) \(25\) \(60\)
    Amount saved (dollars)
20.

Delbert wants to enlarge his class photograph. Its height is \(\dfrac{1}{4}\) of its width. The enlarged photo should have the same shape as the original.

  1. If \(W\) stands for the width of the enlargement, write an expression for its height.
  2. Evaluate the expression to complete the table.
    Width of photo (inches) \(6\) \(10\) \(24\)
    Height of photo (inches)
21.

If the governor vetoes a bill passed by the State Assembly, \(\dfrac{2}{3}\) of the members present must vote for the bill in order to overturn the veto.

  1. If \(p\) stands for the number of Assembly members present, write an expression for the number of votes needed to overturn a veto.
  2. Evaluate the expression to complete the table.
    Members present \(90\) \(96\) \(120\)
    Votes needed
22.

Marla's investment club buys some stock. She will get \(\dfrac{3}{5}\) of the dividends.

  1. If \(D\) stands for the stock dividends, write an expression for Marla's share.
  2. Evaluate the expression to complete the table.
    Stock dividend \(40\) \(85\) \(115\)
    Marla's share \(24\) \(51\) \(69\)

For Problems 23–30, choose a variable for the unknown quantity and translate the phrase into an algebraic expression.

23.

The product of the ticket price and $15

24.

Three times the cost of a light bulb

25.

Three-fifths of the savings account balance

26.

The price of the pizza divided by 6

27.

The weight of the copper in ounces divided by 16

28.

9% of the school buses

29.

$16 less than the cost of the vaccine

30.

The total cost of 32 identical computers

For Problems 31–32, write algebraic expressions in terms of \(x\text{.}\)

31.
  1. Daniel and Lara together made $480. If Daniel made \(x\) dollars, how much did Lara make?
  2. Alix spent $500 on tuition and books. If she spent \(x\) dollars on books, how much was her tuition?
  3. Thirty children signed up for summer camp. If \(x\) boys signed up, how many girls signed up?
32.
  1. Rona spent $15 less than her sister on shoes. If Rona's sister spent \(x\) dollars, how much did Rona spend?
  2. Phoenix had 12 fewer rain days than Boston last year. If Boston had \(x\) rain days, how many rain days did Phoenix have?
  3. Jared scored 18 points lower on his second test than he scored on his first test. If he scored \(x\) points on the first test, what was his score on the second test?

For Problems 33–36, name the variable and write an algebraic expression.

33.

Eggnog is 70% milk. Write an expression for the amount of milk in a container of eggnog.

34.

Errol has saved $1200 for his vacation this year. Write an expression for the average amount he can spend on each day of his vacation.

35.

Garth received 432 fewer votes than his opponent in the election. Write an expression for the number of votes Garth received.

36.

The cost of the conference was $2000 over budget. Write an expression for the cost of the conference.

For Problems 37–40, use the formulas in this Lesson.

37.
  1. Write an equation for the distance \(d\) traveled in \(t\) hours by a small plane flying at 180 miles per hour.
  2. How far will the plane fly in 2 hours? In \(3\dfrac{1}{2}\) hours? In half a day?
38.
  1. A certain pesticide contains 0.02% by volume of a dangerous chemical. Write an equation for the amount of chemical \(C\) that enters the environment in terms of the number of gallons of pesticide used.
  2. How much of the chemical enters the environment if 400 gallons of the pesticide are used? 5000 gallons? 50,000 gallons?
39.
  1. BioTech budgets 8.5% of its revenue for research. Write an equation for the research budget \(B\) in terms of BioTech's revenue \(R\text{.}\)
  2. What is the research budget if BioTech's revenue is $100,000? $500,000? $2,000,000?
40.
  1. Hugo's Auto Shop paid $4000 in expenses this month. Write an equation for their profit \(P\) in terms of their revenue \(R\text{.}\)
  2. What was their profit if their revenue was $10,000? $6500? $2500?