2.1: Introduction to Algebra

Learning Objectives

• Identify an algebraic expression and its parts.
• Evaluate algebraic expressions.
• Use formulas to solve problems in common applications.

Preliminary Definitions

In algebra, letters are used to represent numbers. The letters used to represent these numbers are called variables. Combinations of variables and numbers along with mathematical operations form algebraic expressions, or just expressions.

The following are some examples of expressions with one variable, $x$:

$2x+3\qquad x^{2}-9\qquad 3x^{2}+2x-1\qquad\frac{x-5}{x^{2}-25}$

Terms in an algebraic expression are separated by addition operators, and factors are separated by multiplication operators. The numerical factor of a term is called the coefficient. For example, the algebraic expression $3x^{2}+2x−1$ can be thought of as $3x^{2}+2x+(−1)$ and has three terms. The first term, $3x^{2}$, represents the quantity $3⋅x⋅x$, where $3$ is the coefficient and $x$ is the variable. All of the variable factors, with their exponents, form the variable part of a term. If a term is written without a variable factor, then it is called a constant term. Consider the components of $3x^{2}+2x−1$,

Table $\PageIndex{1}$

Terms	Coefficient	Variable Part
$3x^{2}$	$3$	$x^{2}$
$2x$	$2$	$x$
$-1$	$-1$

The third term in this expression, $−1$, is a constant term because it is written without a variable factor. While a variable represents an unknown quantity and may change, the constant term does not change.

Example $\PageIndex{1}$

List all coefficients and variable parts of each term:

$5x^{2}−4xy−y^{2}$.

Solution:

Think of the third term in this example, $−y^{2}$, as $−1y^{2}$.

Table $\PageIndex{2}$

Terms	Coefficient	Variable Part
$5x^{2}$	$5$	$x^{2}$
$-4xy$	$-4$	$xy$
$-y^{2}$	$-1$	$y^{2}$

Answer:

Coefficients: $\{−4, −1, 5\}$; variable parts: $\{x^{2}, xy, y^{2}\}$

Some terms, such as $y^{2}$ and $−y^{2}$, appear not to have a coefficient. The multiplicative identity property states that $1$ times anything is itself and occurs so often that it is customary to omit this factor and write

$\begin{aligned} 1y^{2}&=y^{2} \\ -1y^{2}&=-y^{2} \end{aligned}$

Therefore, the coefficient of $y^{2}$ is actually $1$ and the coefficient of $−y^{2}$ is $−1$. In addition, you will encounter terms that have variable parts composed of algebraic expressions as factors.

Example $\PageIndex{2}$

List all coefficients and variable parts of each term:

$−3(x+y)^{3}+(x+y)^{2}$

Solution:

This is an expression with two terms:

Table $\PageIndex{3}$

Terms	Coefficient	Variable Part
$-3(x+y)^{3}$	$-3$	$(x+y)^{3}$
$(x+y)^{2}$	$1$	$(x+y)^{2}$

Answer:

Coefficients: $\{−3, 1\}$; variable parts: $\{(x+y)^{3}, (x+y)^{2}\}$

In our study of algebra, we will encounter a wide variety of algebraic expressions. Typically, expressions use the two most common variables, $x$ and $y$. However, expressions may use any letter (or symbol) for a variable, even Greek letters, such as alpha ($α$) and beta ($β$). Some letters and symbols are reserved for constants, such as $π≈3.14159$ and $e≈2.71828$. Since there is only a limited number of letters, you will also use subscripts, $x^{1}, x^{2}, x^{3}, x^{4},…,$ to indicate different variables.

Exercise $\PageIndex{1}$

List all coefficients and variable parts of the expression:

$−5a^{2}+ab−2b^{2}−3$.

Answer

Coefficients: $\{−5, −3, −2, 1\}$; variable parts: $\{a^{2}, ab, b^{2}\}$

Evaluating Algebraic Expressions

Think of an algebraic expression as a generalization of particular arithmetic operations. Performing these operations after substituting given values for variables is called evaluating. In algebra, a variable represents an unknown value. However, if the problem specifically assigns a value to a variable, then you can replace that letter with the given number and evaluate using the order of operations.

Example $\PageIndex{3}$

Evaluate:

1. $2x+3$, where $x=−4$
2. $\frac{2}{3}y$, where $y=9$

Solution:

To avoid common errors, it is a best practice to first replace all variables with parentheses and then replace, or substitute, the given value.

a.

b.

$\begin{aligned} \frac{2}{3}y&=\frac{2}{3}(\:\:) \\ &=\frac{2}{3}(\color{Cerulean}{9}\color{black}{)} \\ &=2\cdot 3\\&=6 \end{aligned}$

Answer:

1. $-5$
2. $6$

If parentheses are not used in part (a) of the previous example, the result is quite different: $2x+3=2−4+4$. Without parentheses, the first operation is subtraction, which leads to an incorrect result.

Example $\PageIndex{4}$

Evaluate:

$−2x−y$, where $x=−5$ and $y=−3$.

Solution:

After substituting the given values for the variables, simplify using the order of operations.

$\begin{aligned} -2x-y&=-2(\:\:)-(\:\:)&\color{Cerulean}{Replace\:variables\:with\:parentheses.} \\ &=-2(\color{Cerulean}{-5}\color{black}{)-(}\color{Cerulean}{-3}\color{black}{)}&\color{Cerulean}{Substitute\:values\:for\:x\:and\:y.} \\ &=10+3&\color{Cerulean}{Simplify.} \\&=13 \end{aligned}$

Answer:

$13$

Example $\PageIndex{5}$

Evaluate:

$9a^{2}−b^{2}$, where $a=2$ and $b=−5$.

Solution:

Answer:

$11$

Example $\PageIndex{6}$

Evaluate:

$−x^{2}−4x+1$, where $x=−\frac{1}{2}$.

Solution:

$\begin{aligned} -x^{2}-4x+1&=-(\:\:)^{2}-4(\:\:)+1 &\color{Cerulean}{Apply\:the\:exponent\:first.} \\ &=\color{black}{-\left( \color{Cerulean}{-\frac{1}{2}}\right)^{2}-4\left(\color{Cerulean}{-\frac{1}{2}}\right)+1} &\color{Cerulean}{(-\frac{1}{2})^{2}=(-\frac{1}{2})(-\frac{1}{2})=\frac{1}{4}} \\&=-\left(\frac{1}{4} \right)+2+1 \\ &=-\frac{1}{4}+3 \\ &=-\frac{1}{4}+\frac{12}{4} \\ &=\frac{11}{4} \end{aligned}$

Answer:

$\frac{11}{4}$

The answer to the previous example is $\frac{11}{4}$, which can be written as a mixed number $2\frac{3}{4}$. In algebra, improper fractions are generally preferred. Unless the original problem has mixed numbers in it, or it is an answer to a real-world application, solutions will be expressed as reduced improper fractions.

Example $\PageIndex{7}$

Evaluate:

$(3x−2)(x−7)$, where $x=\frac{2}{3}$

Solution:

The order of operations requires us to perform the operations within the parentheses first.

$\begin{aligned} (3x-2)(x-7)&=(3(\:\:)-2)((\:\:)-7) \\ &=\color{black}{\left(3\left (\color{Cerulean}{\frac{2}{3}}\right)-2\right)\left(\left(\color{Cerulean}{\frac{2}{3}}\right)-7\right)} \\ &=(2-2)\left(\frac{2}{3}-\frac{21}{3} \right) \\ &=(0)\left(-\frac{19}{3} \right) \\ &=0 \end{aligned}$

Answer:

$0$

Example $\PageIndex{8}$

Evaluate:

$b^{2}−4ac$, where $a=−1, b=−3,$ and $c=2$.

Solution:

The expression $b^{2}−4ac$ is called the discriminant; it is an essential quantity seen later in our study of algebra.

$\begin{aligned} b^{2}-4ac&=(\:\:)^{2}-4(\:\:)(\:\:)&\color{Cerulean}{Exponents\:first,\:then\:multiplication} \\ &=(\color{Cerulean}{-3}\color{black}{)^{2}-4(}\color{Cerulean}{-1}\color{black}{)(}\color{Cerulean}{2}\color{black}{)} &\color{Cerulean}{(-3)^{2}=(-3)(-3)=+9} \\ &=9+4(2) \\ &=9+8 \\ &=17 \end{aligned}$

Answer:

$17$

Exercise $\PageIndex{2}$

Evaluate $a^{3}−b^{3}$, where $a=2$ and $b=−3$.

Answer

$35$

Using Formulas

The main difference between algebra and arithmetic is the organized use of variables. This idea leads to reusable formulas, which are mathematical models using algebraic expressions to describe common applications. For example, the area of a rectangle is modeled by the formula:

$$A=l\cdot w$$

In this equation, variables are used to describe the relationship between the area of a rectangle and the length of its sides. The area is the product of the length and width of the rectangle. If the length of a rectangle measures $3$ meters and the width measures $2$ meters, then the area can be calculated using the formula as follows:

$\begin{aligned} A&=l\cdot w\\ &=3\text{m}\cdot (2\text{m}) \\ &=6\text{ square meters}(\text{m}^{2}) \end{aligned}$

Example $\PageIndex{9}$

The cost of a daily truck rental is $$48.00$ plus an additional $$0.45$ for every mile driven. This cost in dollars can be modeled by the formula $cost=0.45x+48$, where $x$ represents the number of miles driven in one day. Use this formula to calculate the cost to rent the truck for a day and drive it $120$ miles.

Solution:

Use the formula to find the cost when the number of miles $x=120$.

$\color{Cerulean}{cost}\color{black}{=0.45x+48}$

Substitute $120$ into the given formula for $x$ and then simplify.

$\begin{aligned} \color{Cerulean}{cost}&=0.45(\:\:)+48 \\ &=0.45(\color{Cerulean}{120}\color{black}{)+48} \\ &=54+48 \\ &=102 \end{aligned}$

Answer:

The rental costs $$102$.

Uniform motion is modeled by the formula $D=rt$, which expresses distance $D$ in terms of the average rate $r$, or speed, and the time $t$ traveled at that rate. This formula, $D=rt$, is used often and is read “distance equals rate times time.”

Example $\PageIndex{10}$

Jim’s road trip takes $2\frac{1}{2}$ hours at an average speed of $66$ miles per hour. How far does he travel?

Solution:

Substitute the appropriate values into the formula and then simplify.

$\begin{aligned} D&=r\cdot t \\ &=\color{black}{\left(\color{Cerulean}{66}\:\frac{mi}{hr} \right)\cdot\left(\color{Cerulean}{2\frac{1}{2}\: hr} \right)} \\ &=\frac{66}{1}\cdot\frac{5}{2}mi \\ &=33\cdot 5mi \\ &=165 mi \end{aligned}$

Answer:

Jim travels $165$ miles.

The volume in cubic units of a rectangular box is given by the formula $V=lwh$, where $l$ represents the length, $w$ represents the width, and $h$ represents the height.

Figure $\PageIndex{1}$

Example $\PageIndex{11}$

A wooden box is $1$ foot in length, $5$ inches wide, and $6$ inches high. Find the volume of the box in cubic inches.

Solution:

Take care to ensure that all the units are consistent and use $12$ inches for the length instead of $1$ foot.

$\begin{aligned} V&=lwh \\ V&=(\:\:)(\:\:)(\:\:) \\ &=(\color{Cerulean}{12in}\color{black}{)(}\color{Cerulean}{5in}\color{black}{)(}\color{Cerulean}{6in}\color{black}{)} \\ &=360\:\text{cubic inches }(\text{in}^{3}) \end{aligned}$

Answer:

The volume of the box is $360$ cubic inches.

Simple interest $I$ is given by the formula $I=prt$, where $p$ represents the principal amount invested at an annual interest rate $r$ for $t$ years.

Example $\PageIndex{12}$

Calculate the simple interest earned on a 2-year investment of $$1,250$ at an annual interest rate of $3\frac{3}{4}$%.

Solution:

Convert $3\frac{3}{4}$% to a decimal number before using it in the formula.

$r=3\frac{3}{4}$%$=3.75$%$=0.0375$

Use this value for $r$ and the fact that $p =$ $$1,250$ and $t =$ 2 years to calculate the simple interest.

$\begin{aligned} I&=prt \\ &=(\color{Cerulean}{1,250}\color{black}{)(}\color{Cerulean}{0.0375}\color{black}{)(}\color{Cerulean}{2}\color{black}{)}\\&=93.75 \end{aligned}$

Answer:

The simple interest earned is $$93.75$.

Exercise $\PageIndex{3}$

The perimeter of a rectangle is given by the formula $P=2l+2w$, where $l$ represents the length and $w$ represents the width. Use the formula to calculate the perimeter of a rectangle with a length of $5$ feet and a width of $2\frac{1}{2}$ feet.

Answer

$15$ feet

Key Takeaways

• Think of algebraic expressions as generalizations of common arithmetic operations that are formed by combining numbers, variables, and mathematical operations.
• It is customary to omit the coefficient if it is $1$, as in $x^{2}=1x^{2}$.
• To avoid common errors when evaluating, it is a best practice to replace all variables with parentheses and then substitute the appropriate values.
• The use of algebraic expressions allows us to create useful and reusable formulas that model common applications.

Exercise $\PageIndex{4}$ Definitions

List all of the coefficients and variable parts of the following expressions.

1. $4x−1$
2. $–7x^{2}−2x+1$
3. $−x^{2}+5x−3$
4. $3x^{2}y^{2}−\frac{2}{3}xy+7$
5. $\frac{1}{3}y^{2}−\frac{1}{2}y+\frac{5}{7}$
6. $−4a^{2}b+5ab^{2}−ab+1$
7. $2(a+b)^{3}−3(a+b)^{5}$
8. $5(x+2)^{2}−2(x+2)−7$
9. $m^{2}n−mn^{2}+10mn−27$
10. $x^{4}−2x^{3}−3x^{2}−4x−1$

Answer

1. Coefficients: $\{−1, 4\}$; variable parts: $\{x\}$

3. Coefficients: $\{−3, −1, 5\}$; variable parts: $\{x^{2}, x\}$

5. Coefficients: $\{−\frac{1}{2}, \frac{1}{3}, \frac{5}{7}\}$; variable parts: $\{y^{2}, y\}$

7. Coefficients: $\{−3, 2\}$; variable parts: $\{(a+b)^{3},(a+b)^{5}\}$

9. Coefficients: $\{−27, −1, 1, 10\}$; variable parts: $\{m^{2}n, mn^{2}, mn\}$

Exercise $\PageIndex{5}$ Evaluating Algebraic Expressions

Evaluate.

1. $x+3$, where $x=−4$
2. $2x−3$, where $x=−3$
3. $−5x+20$, where $x=4$
4. $-5y$, where $y=−1$
5. $\frac{3}{4}a$, where $a=32$
6. $2(a−4)$, where $a=−1$
7. $−10(5−z)$, where $z=14$
8. $5y−1$, where $y=−\frac{1}{5}$
9. $−2a+1$, where $a=−\frac{1}{3}$
10. $4x+3$, where $x=\frac{3}{16}$
11. $−x+\frac{1}{2}$, where $x=−2$
12. $\frac{2}{3}x−\frac{1}{2}$, where $x=−\frac{1}{4}$

Answer

1. $−1$

3. $0$

5. $24$

7. $90$

9. $\frac{5}{3}$

11. $\frac{5}{2}$

Exercise $\PageIndex{6}$ Evaluating Algebraic Expressions

For each problem below, evaluate $b^{2}−4ac$, given the following values for $a, b,$ and $c$.

1. $a=1, b=2, c=3$
2. $a=3, b=–4, c=–1$
3. $a=–6, b=0, c=–2$
4. $a=\frac{1}{2}, b=1, c=\frac{2}{3}$
5. $a=−3, b=−\frac{1}{2}, c=\frac{1}{9}$
6. $a=−\frac{1}{3}, b=−\frac{2}{3}, c=0$

Answer

1. $-8$

3. $-48$

5. $\frac{19}{12}$

Exercise $\PageIndex{7}$ Evaluating Algebraic Expressions

Evaluate.

1. $−4xy^{2}$, where $x=−3$ and $y=2$
2. $\frac{5}{8}x^{2}y$, where $x=−1$ and $y=16$
3. $a^{2}−b^{2}$, where $a=2$ and $b=3$
4. $a^{2}−b^{2}$, where $a=−1$ and $b=−2$
5. $x^{2}−y^{2}$, where $x=\frac{1}{2}$ and $y=−\frac{1}{2}$
6. $3x^{2}−5x+1$, where $x=−3$
7. $y^{2}−y−6$, where $y=0$
8. $1−y^{2}$, where $y=−\frac{1}{2}$
9. $(x+3)(x−2)$, where $x=−4$
10. $(y−5)(y+6)$, where $y=5$
11. $3(α−β)+4$, where $α=−1$ and $β=6$
12. $3α^{2}−β^{2}$, where $α=2$ and $β=−3$
13. Evaluate $4(x+h)$, given $x=5$ and $h=0.01$.
14. Evaluate $−2(x+h)+3$, given $x=3$ and $h=0.1$.
15. Evaluate $2(x+h)^{2}−5(x+h)+3$, given $x=2$ and $h=0.1$.
16. Evaluate $3(x+h)^{2}+2(x+h)−1$, given $x=1$ and $h=0.01$.

Answer

1. $48$

3. $−5$

5. $0$

7. $−6$

9. $6$

11. $−17$

13. $20.04$

15. $1.32$

Exercise $\PageIndex{8}$ Using Formulas

Convert the following temperatures to degrees Celsius given $C=\frac{5}{9}(F−32)$, where $F$ represents degrees Fahrenheit.

1. $86°F$
2. $95°F$
3. $−13°F$
4. $14°F$
5. $32°F$
6. $0°F$

Answer

1. $30°C$

3. $−25°C$

5. $0°C$

Exercise $\PageIndex{9}$ Using Formulas

Given the base and height of a triangle, calculate the area. $(A=\frac{1}{2}bh)$.

1. $b=25$ centimeters and $h=10$ centimeters
2. $b=40$ inches and $h=6$ inches
3. $b=\frac{1}{2}$ foot and $h=2$ feet
4. $b=\frac{3}{4}$ inches and $h=\frac{5}{8}$ inches

Answer

1. $125$ square centimeters

3. $\frac{1}{2}$ square feet

Exercise $\PageIndex{10}$ Using Formulas

1. A certain cellular phone plan charges $$23.00$ per month plus $$0.09$ for each minute of usage. The monthly charge is given by the formula monthly $charge=0.09x+23$, where $x$ represents the number of minutes of usage per month. What is the charge for a month with $5$ hours of usage?
2. A taxi service charges $$3.75$ plus $$1.15$ per mile given by the formula $charge=1.15x+3.75$, where $x$ represents the number of miles driven. What is the charge for a $17$-mile ride?
3. If a calculator is sold for $$14.95$, then the revenue in dollars, $R$, generated by this item is given by the formula $R=14.95q$, where $q$ represents the number of calculators sold. Use the formula to determine the revenue generated by this item if $35$ calculators are sold.
4. Yearly subscriptions to a tutoring website can be sold for $$49.95$. The revenue in dollars, $R$, generated by subscription sales is given by the formula $R=49.95q$, where $q$ represents the number of yearly subscriptions sold. Use the formula to calculate the revenue generated by selling $250$ subscriptions.
5. The cost of producing pens with the company logo printed on them consists of a onetime setup fee of $$175$ plus $$0.85$ for each pen produced. This cost can be calculated using the formula $C=175+0.85q$, where $q$ represents the number of pens produced. Use the formula to calculate the cost of producing $2,000$ pens.
6. The cost of producing a subscription website consists of an initial programming and setup fee of $$4,500$ plus a monthly Web hosting fee of $$29.95$. The cost of creating and hosting the website can be calculated using the formula $C=4500+29.95n$, where $n$ represents the number of months the website is hosted. How much will it cost to set up and host the website for 1 year?
7. The perimeter of a rectangle is given by the formula $P=2l+2w$, where $l$ represents the length and $w$ represents the width. What is the perimeter of a fenced-in rectangular yard measuring $70$ feet by $100$ feet?
8. Calculate the perimeter of an $8$-by-$10$-inch picture.
9. Calculate the perimeter of a room that measures $12$ feet by $18$ feet.
10. A computer monitor measures $57.3$ centimeters in length and $40.9$ centimeters high. Calculate the perimeter.
11. The formula for the area of a rectangle in square units is given by $A=l⋅w$, where $l$ represents the length and $w$ represents the width. Use this formula to calculate the area of a rectangle with length $12$ centimeters and width $3$ centimeters.
12. Calculate the area of an $8$-by-$12$-inch picture.
13. Calculate the area of a room that measures $12$ feet by $18$ feet.
14. A computer monitor measures $57.3$ centimeters in length and $40.9$ centimeters in height. Calculate the total area of the screen.
15. A concrete slab is poured in the shape of a rectangle for a shed measuring $8$ feet by $10$ feet. Determine the area and perimeter of the slab.
16. Each side of a square deck measures $8$ feet. Determine the area and perimeter of the deck.
17. The volume of a rectangular solid is given by $V=lwh$, where $l$ represents the length, $w$ represents the width, and $h$ is the height of the solid. Find the volume of a rectangular solid if the length is $2$ inches, the width is $3$ inches, and the height is $4$ inches.
18. If a trunk measures $3$ feet by $2$ feet and is $2\frac{1}{2}$ feet tall, then what is the volume of the trunk?
19. The interior of an industrial freezer measures $3$ feet wide by $3$ feet deep and $4$ feet high. What is the volume of the freezer?
20. A laptop case measures $1$ feet $2$ inches by $10$ inches by $2$ inches. What is the volume of the case?
21. If the trip from Fresno to Sacramento can be made by car in $2\frac{1}{2}$ hours at an average speed of $67$ miles per hour, then how far is Sacramento from Fresno?
22. A high-speed train averages $170$ miles per hour. How far can it travel in $1\frac{1}{2}$ hours?
23. A jumbo jet can cruises at an average speed of $550$ miles per hour. How far can it travel in $4$ hours?
24. A fighter jet reaches a top speed of $1,316$ miles per hour. How far will the jet travel if it can sustain this speed for $15$ minutes?
25. The Hubble Space Telescope is in low earth orbit traveling at an average speed of $16,950$ miles per hour. What distance does it travel in $1\frac{1}{2}$ hours?
26. Earth orbits the sun a speed of about $66,600$ miles per hour. How far does earth travel around the sun in 1 day?
27. Calculate the simple interest earned on a $$2,500$ investment at $3$% annual interest rate for 4 years.
28. Calculate the simple interest earned on a $$1,000$ investment at $5$% annual interest rate for 20 years.
29. How much simple interest is earned on a $$3,200$ investment at a $2.4$% annual interest for 1 year?
30. How much simple interest is earned on a $$500$ investment at a $5.9$% annual interest rate for 3 years?
31. Calculate the simple interest earned on a $$10,500$ investment at a $4\frac{1}{4}$% annual interest rate for 4 years.
32. Calculate the simple interest earned on a $$6,250$ investment at a $6\frac{3}{4}$% annual interest rate for 1 year.

Answer

1. $$50$

3. $$523.25$

5. $$1,875.00$

7. $340$ feet

9. $60$ feet

11. $36$ square centimeters

13. $216$ square feet

15. Area: $80$ square feet; Perimeter: $36$ feet

17. $24$ cubic inches

19. $36$ cubic feet

21. $167.5$ miles

23. $2,200$ miles

25. $25,425$ miles

27. $$300$

29. $$76.80$

31. $$1,785$

Exercise $\PageIndex{11}$ Discussion Board Topics

1. Research and discuss the history of the symbols for addition ($+$) and subtraction ($−$).
2. What are mathematical models and why are they useful in everyday life?
3. Find and post a useful formula. Demonstrate its use with some values.
4. Discuss the history and importance of the variable. How can you denote a variable when you run out of letters?
5. Find and post a useful resource describing the Greek alphabet.

Answer

1. Answers may vary

3. Answers may vary

5. Answers may vary