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Section 2.1 Variables and Evaluating Expressions

To move past arithmetic to algebra, we begin working with variables. Any combination of numbers and variables using mathematical operations is called an algebraic expression. Some expressions are simple, and some are complicated. Some expressions are abstract, whereas some have context and meaning. One example of a simple algebraic expression with context is β€œ220βˆ’a,” which has one variable, a, and is the expression for the maximum heart rate of a person who is a years old.

In this section, we'll focus on variables and expressions. In Section 2.2 we'll continue with a focus on geometry formulas. In the remainder of this chapter, we'll focus on mathematical equations and inequalities which are also very important in algebra.

Figure 2.1.1 Alternative Video Lesson

Subsection 2.1.1 Introduction to Variables

When we want to represent an unknown or changing numerical quantity, we use a variable to do so. For example, if you'd like to discuss the gas mileage of various cars, you could let the symbol β€œg” represent a car's gas mileage. The mileage might be 25β€―mpg, 30β€―mpg, or some other quantity. If we agree to use mpg for the units of measure, g might be a place holder for 25, 30, or some other number. Since we are using a variable and not a specific number, we can discuss gas mileage for Honda Civics at the same time we discuss gas mileage for Ford Explorers.

When variables stand for actual physical quantities, it's good to use letters that clearly correspond to the quantity they represent. For example, it's wise to use g to represent gas mileage. This helps the people who might read your work in the future to understand it better.

It is also important to identify what unit of measurement goes along with each variable you use, and clearly tell your reader this. For example, suppose you are working with g=25. A car whose gas mileage is 25β€―mpg is very different from a car whose gas mileage is 25β€―kpg (kilometers per gallon). So it would be important to tell readers that g represents gas mileage in miles per gallon.

Checkpoint 2.1.2

Unless an algebra problem specifies which letter(s) to use, we may choose which letter(s) to use for our variable(s). However without any context to a problem, x, y, and z are the most common letters used as variables, and you may see these variables (especially x) a lot.

Also note that the units we use are often determined indirectly by other information given in an algebra problem. For example, if we're told that a car has used so many gallons of gas after traveling so many miles, then it suggests we should measure gas mileage in mpg.

Subsection 2.1.2 Algebraic Expressions

An algebraic expression is any combination of variables and numbers using arithmetic operations. The following are all examples of algebraic expressions:

x+12β„“+2wxy+1nRT

Note that this definition of β€œalgebraic expression” does not include anything with signs like these in them: =, <, ≀, etc.

Example 2.1.3

The expression:

59(Fβˆ’32)

can be used to convert from 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 calculating the result. In this case, we can replace F with a specific number.

Let's convert the temperature 89β€―Β°F to the Celsius scale by evaluating the expression.

59(Fβˆ’32)=59(89βˆ’32)=59(57)=2859β‰ˆ31.67

This shows us that 89β€―Β°F is equivalent to approximately 31.67β€―Β°C.

Warning 2.1.4 Evaluating Versus Solving

The steps in Example 2.1.3 are not considered β€œsolving” anything. β€œSolving” is a word you might be tempted to use, because in some sense the steps from Example 2.1.3 are β€œfinding an answer.” There is a special meaning in algebra for words like β€œsolve” and β€œsolution” that will come soon. Instead, when we substitute a value and compute the result, the proper vocabulary is β€œevaluating an expression.”

Checkpoint 2.1.5

Try evaluating the temperature expression for yourself.

Example 2.1.6 Target heart rate

According to the American Heart Association, a person's maximum heart rate, in beats per minute (bpm), is given by 220βˆ’a, where a is their age in years.

  1. Determine the maximum heart rate for someone who is 31 years old.

  2. A person's target heart rate for moderate exercise is 50% to 70% of their maximum heart rate. If they want to reach 65% of their maximum heart rate during moderate exercise, we'd use the expression 0.65(220βˆ’a), where a is their age in years. Determine the target heart rate at this 65% level for someone who is 31 years old.

Explanation

Both of these parts ask us to evaluate an expression.

  1. Since \(a\) is defined to be age in years, we will evaluate this expression by substituting \(a\) with \(31\text{:}\)

    \begin{align*} 220-a \amp= 220-\substitute{31}\\ \amp= 189 \end{align*}

    This tells us that the maximum heart rate for someone who is \(31\) years old is \(189\) bpm.

  2. We'll again substitute \(a\) with \(31\text{,}\) but this time using the target heart rate expression:

    \begin{align*} 0.65(220-a) \amp= 0.65(220-\substitute{31})\\ \amp= 0.65(189)\\ \amp=122.85 \end{align*}

    This tells us that the target heart rate for someone who is \(31\) years old undertaking moderate exercise is \(122.85\) bpm.

Checkpoint 2.1.7
Checkpoint 2.1.8 Rising Rents

Subsection 2.1.3 Evaluating Expressions with Exponents, Absolute Value, and Radicals

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

Example 2.1.9 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 30β€―m to four significant digits.

Explanation

Using \(d=30\text{,}\) we find:

\begin{align*} \sqrt{9.8d} \amp= \sqrt{9.8(\substitute{30})}\\ \amp=\sqrt{294}\\ \amp\approx \overbrace{17.14}^{\text{four}}6428\ldots \end{align*}

The speed of tsunami with a depth of 30β€―m is about 17.15β€―m⁄s.

Up to now, we have been evaluating expressions, but we can evaluate formulas in the same way. A formula usually has a single variable that represents the output of an expression. For example, the expression for a person's maximum heart rate in beats per minute, 220βˆ’a, can be written as the formula, H=220βˆ’a. When we substitute a value for a we are evaluating the formula. Even though we have an equation, we are not solving yet. That will come soon.

Checkpoint 2.1.10 Tent Height
Checkpoint 2.1.11 Mortgage Payments
Warning 2.1.12 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. Let's look at some examples.

Example 2.1.13

Evaluate x2 if x=βˆ’2.

We substitute:

x2=(βˆ’2)2=4

If we don't use parentheses, we would have:

x2=βˆ’22incorrect!=βˆ’4

The original expression takes x and squares it. With βˆ’22=βˆ’4, the number βˆ’2 is not being squared. Since the exponent has higher priority than the negation, it's just the number 2 that is being squared. With (βˆ’2)2=4 the number βˆ’2 is being squared, which is what we would want given the expression x2.

So it is wise to always use some parentheses when substituting in any negative number.

Checkpoint 2.1.14

Subsection 2.1.4 Exercises

Evaluating Expressions
1

Evaluate xβˆ’10 for x=6.

2

Evaluate x+4 for x=9.

3

Evaluate βˆ’4βˆ’x for x=βˆ’10.

4

Evaluate 10βˆ’x for x=βˆ’8.

5

Evaluate 3x+6 for x=βˆ’5.

6

Evaluate βˆ’5xβˆ’6 for x=βˆ’3.

7

Evaluate βˆ’7c for c=9.

8

Evaluate βˆ’2B for B=2.

9

Evaluate the expression r2:

  1. When r=3, r2=

  2. When r=βˆ’5, r2=

10

Evaluate the expression t2:

  1. When t=9, t2=

  2. When t=βˆ’9, t2=

11

Evaluate the expression t3:

  1. When t=4, t3=

  2. When t=βˆ’3, t3=

12

Evaluate the expression x3:

  1. When x=2, x3=

  2. When x=βˆ’4, x3=

13

Evaluate the following expressions.

  1. Evaluate 5x2 when x=2. 5x2=

  2. Evaluate (5x)2 when x=2. (5x)2=

14

Evaluate the following expressions.

  1. Evaluate 3x2 when x=2. 3x2=

  2. Evaluate (3x)2 when x=2. (3x)2=

15

Evaluate βˆ’(y+2) for y=βˆ’7.

16

Evaluate βˆ’7(y+9) for y=6.

17

Evaluate 9rβˆ’53r for r=βˆ’1.

18

Evaluate 3rβˆ’73r for r=βˆ’8.

19

Evaluate βˆ’9B+9A for B=βˆ’10 and A=βˆ’6.

20

Evaluate βˆ’2Cβˆ’c for C=βˆ’9 and c=5.

21

Evaluate βˆ’5xβˆ’4a for x=7 and a=βˆ’3.

22

Evaluate βˆ’5yβˆ’8B for y=9 and B=βˆ’5.

23

Evaluate βˆ’7t+9cβˆ’37tβˆ’2c for t=βˆ’4 and c=9.

24

Evaluate βˆ’2aβˆ’B+9βˆ’8a+7B for a=3 and B=βˆ’7.

25

Evaluate the expression 17(x+2)2βˆ’7 when x=βˆ’9.

26

Evaluate the expression 14(x+3)2βˆ’4 when x=βˆ’7.

27

Evaluate the expression 12h(B+b) when h=10, B=8, b=7.

28

Evaluate the expression 12h(B+b) when h=12, B=6, b=5.

29

Evaluate the expression βˆ’16t2+64t+128 when t=3.

30

Evaluate the expression βˆ’16t2+64t+128 when t=βˆ’5.

31

Evaluate the following expressions.

  1. Evaluate x2r3 when x=βˆ’3 and r=βˆ’1.

    x2r3=

  2. Evaluate x3r2 when x=βˆ’3 and r=βˆ’1.

    x3r2=

32

Evaluate the following expressions.

  1. Evaluate x2y3 when x=βˆ’1 and y=βˆ’2.

    x2y3=

  2. Evaluate x3y2 when x=βˆ’1 and y=βˆ’2.

    x3y2=

33

Evaluate the following expressions.

  1. Evaluate (βˆ’3y)2 when y=βˆ’1.

    (βˆ’3y)2=

  2. Evaluate (βˆ’3y)3 when y=βˆ’1.

    (βˆ’3y)3=

34

Evaluate the following expressions.

  1. Evaluate (βˆ’y)2 when y=βˆ’2.

    (βˆ’y)2=

  2. Evaluate (βˆ’y)3 when y=βˆ’2.

    (βˆ’y)3=

35

Evaluate each algebraic expression for the given value(s):

y3+xβˆ’4|2xβˆ’y|, for x=104 and y=βˆ’4:

36

Evaluate each algebraic expression for the given value(s):

y3+xβˆ’4|5xβˆ’y|, for x=29 and y=7:

37

Evaluate each algebraic expression for the given value(s):

xyβˆ’yx, for x=81 and y=3:

38

Evaluate each algebraic expression for the given value(s):

xyβˆ’yx, for x=100 and y=βˆ’6:

39

Evaluate

y2βˆ’y1x2βˆ’x1

for x1=βˆ’20, x2=13, y1=19, and y2=βˆ’1:

40

Evaluate

y2βˆ’y1x2βˆ’x1

for x1=βˆ’15, x2=βˆ’2, y1=βˆ’5, and y2=βˆ’12:

41

Evaluate

(x2βˆ’x1)2+(y2βˆ’y1)2

for x1=8, x2=4, y1=8, and y2=11:

42

Evaluate

(x2βˆ’x1)2+(y2βˆ’y1)2

for x1=βˆ’2, x2=4, y1=βˆ’8, and y2=βˆ’16:

43

Evaluate the algebraic expression 3a+b for a=57 and b=49.

44

Evaluate the algebraic expression βˆ’8a+b for a=65 and b=12.

45

Evaluate each algebraic expression for the given value(s):

5+5|yβˆ’x|x+5y, for x=6 and y=βˆ’3:

46

Evaluate each algebraic expression for the given value(s):

4+5|yβˆ’x|x+3y, for x=11 and y=βˆ’3:

47

To convert a temperature measured in degrees Fahrenheit to degrees Celsius, there is a formula:

C=59(Fβˆ’32)

where C represents the temperature in degrees Celsius and F represents the temperature in degrees Fahrenheit.

If a temperature is 113∘F, what is that temperature measured in Celsius?

48

To convert a temperature measured in degrees Fahrenheit to degrees Celsius, there is a formula:

C=59(Fβˆ’32)

where C represents the temperature in degrees Celsius and F represents the temperature in degrees Fahrenheit.

If a temperature is 5∘F, what is that temperature measured in Celsius?

49

To convert a temperature measured in degrees Fahrenheit to degrees Celsius, there is a formula:

C=59(Fβˆ’32)

where C represents the temperature in degrees Celsius and F represents the temperature in degrees Fahrenheit.

If a temperature is 14∘F, what is that temperature measured in Celsius?

50

To convert a temperature measured in degrees Fahrenheit to degrees Celsius, there is a formula:

C=59(Fβˆ’32)

where C represents the temperature in degrees Celsius and F represents the temperature in degrees Fahrenheit.

If a temperature is 122∘F, what is that temperature measured in Celsius?

51

A formula for converting meters into feet is

F=3.28M

where M is a number of meters, and F is the corresponding number of feet.

Use the formula to find the number of feet that corresponds to fourteen meters.

feet corresponds to fourteen meters.

52

A formula for converting meters into feet is

F=3.28M

where M is a number of meters, and F is the corresponding number of feet.

Use the formula to find the number of feet that corresponds to eight meters.

feet corresponds to eight meters.

53

The formula

y=12at2+v0t+y0

gives the vertical position of an object, at time t, thrown with an initial velocity v0, from an initial position y0 in a place where the acceleration of gravity is a. The acceleration of gravity on earth is βˆ’9.8 (m(s2. It is negative, because we consider the upward direction as positive in this situation, and gravity pulls down.

What is the height of a baseball thrown with an initial velocity of v0=78 (m(s, from an initial position of y0=97 m, and at time t=14 s?

Fourteen seconds after the baseball was thrown, it was high in the air.

54

The formula

y=12at2+v0t+y0

gives the vertical position of an object, at time t, thrown with an initial velocity v0, from an initial position y0 in a place where the acceleration of gravity is a. The acceleration of gravity on earth is βˆ’9.8 (m(s2. It is negative, because we consider the upward direction as positive in this situation, and gravity pulls down.

What is the height of a baseball thrown with an initial velocity of v0=84 (m(s, from an initial position of y0=80 m, and at time t=5 s?

Five seconds after the baseball was thrown, it was high in the air.

55

The percentage of births in the U.S. delivered via C-section can be given by the following formula for the years since 1996:

p=0.8(yβˆ’1996)+21

In this formula y is a year after 1996 and p is the percentage of births delivered via C-section for that year.

What percentage of births in the U.S. were delivered via C-section in the year 2010?

of births in the U.S. were delivered via C-section in the year 2010.

56

The percentage of births in the U.S. delivered via C-section can be given by the following formula for the years since 1996:

p=0.8(yβˆ’1996)+21

In this formula y is a year after 1996 and p is the percentage of births delivered via C-section for that year.

What percentage of births in the U.S. were delivered via C-section in the year 2012?

of births in the U.S. were delivered via C-section in the year 2012.

57

Target heart rate for moderate exercise is 50% to 70% of maximum heart rate. If we want to represent a certain percent of an individual’s maximum heart rate, we’d use the formula

rate=p(220βˆ’a)

where p is the percent, and a is age in years.

Determine the target heart rate at 51% level for someone who is 43 years old. Round your answer to an integer.

The target heart rate at 51% level for someone who is 43 years old is beats per minute.

58

Target heart rate for moderate exercise is 50% to 70% of maximum heart rate. If we want to represent a certain percent of an individual’s maximum heart rate, we’d use the formula

rate=p(220βˆ’a)

where p is the percent, and a is age in years.

Determine the target heart rate at 53% level for someone who is 22 years old. Round your answer to an integer.

The target heart rate at 53% level for someone who is 22 years old is beats per minute.

59

The diagonal length (D) of a rectangle with side lengths L and W is given by:

D=L2+W2

Determine the diagonal length of rectangles with L=6 ft and W=8 ft.

The diagonal length of rectangles with L=6 ft and W=8 ft is .

60

The diagonal length (D) of a rectangle with side lengths L and W is given by:

D=L2+W2

Determine the diagonal length of rectangles with L=9 ft and W=12 ft.

The diagonal length of rectangles with L=9 ft and W=12 ft is .

61

The height inside a camping tent when you are d feet from the edge of the tent is given by

h=βˆ’1.1|dβˆ’5.4|+6

where h stands for height in feet.

Determine the height when you are:

  1. 6.5 ft from the edge.

    The height inside a camping tent when you 6.5 ft from the edge of the tent is

  2. 4.4 ft from the edge.

    The height inside a camping tent when you 4.4 ft from the edge of the tent is

62

The height inside a camping tent when you are d feet from the edge of the tent is given by

h=βˆ’0.6|dβˆ’5.8|+5.5

where h stands for height in feet.

Determine the height when you are:

  1. 9.7 ft from the edge.

    The height inside a camping tent when you 9.7 ft from the edge of the tent is

  2. 2.9 ft from the edge.

    The height inside a camping tent when you 2.9 ft from the edge of the tent is