## Section1.2Algebraic Expressions

### SubsectionWriting 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.

###### Example1.6.

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.

###### 1.

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

### SubsectionSums

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.

###### Example1.7.

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.

If $a$ and $b$ are numbers, then

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

### SubsectionProducts

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:

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*}
###### Example1.8.

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 $z$”.

###### 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 .

### SubsectionDifferences

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.

###### Caution1.9.

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.

### SubsectionQuotients

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.

###### 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 $3 \div 12 = \dfrac{1}{4}\text{.}$

###### Example1.10.

Write an algebraic expression for each phrase.

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

###### 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 .

### SubsectionEvaluating 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.

###### Example1.11.

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 $R=540\text{.}$

### SubsectionSome 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*}
###### Example1.12.

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*}
###### Example1.13.

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*} ###### Example1.14. 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*}

###### 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

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.

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?