**Learning Objectives**

In this section, you will:

- Isolate a given variable on one side of an equation.
- Factor/combine a given variable from/with all terms that have it.
- Divide both sides of an equation to solve for the given variable.
- Reduce careless errors during the process of solving for a given variable.

**Let’s Get Started by Using the Distance, Rate, and Time Formula**

One formula you will use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant rate. Rate is an equivalent word for “speed.” The basic idea of rate may already familiar to you. Do you know what distance you travel if you drive at a steady rate of 60 miles per hour for 2 hours? (This might happen if you use your car’s cruise control while driving on the highway.) If you said 120 miles, you already know how to use this formula!

**Distance, Rate, and Time**

For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula:

[latex]\begin{array}{ccccccccc}d=rt\hfill & & & \hfill \text{where}\hfill & & & \hfill d& =\hfill & \text{distance}\hfill \\ & & & & & & \hfill r& =\hfill & \text{rate}\hfill \\ & & & & & & \hfill t& =\hfill & \text{time}\hfill \end{array}[/latex]

**How To**

Solve an application (with a formula).

**Read**the problem. Make sure all the words and ideas are understood.**Identify**what we are looking for.**Name**what we are looking for. Choose a variable to represent that quantity.**Translate**into an equation. Write the appropriate formula for the situation. Substitute in the given information.**Solve**the equation using good algebra techniques.**Check**the answer in the problem and make sure it makes sense.**Answer**the question with a complete sentence.

**EXAMPLE 1**

**Creating a Mini-Chart to Summarize the Information in the Problem**

Jamal rides his bike at a uniform rate of 12 miles per hour for [latex]3\frac{1}{2}[/latex] hours. What distance has he traveled?

## Show/Hide Solution

#### Solution

Step 1. Read the problem. |
||

Step 2. Identify what you are looking for. |
distance traveled | |

Step 3. Name. Choose a variable to represent it. |
Let [latex]d[/latex] = distance. | |

Step 4. Translate: Write the appropriate formula. |
[latex]d=rt[/latex] | |

Substitute in the given information. | [latex]d=12\cdot 3\frac{1}{2}[/latex] | |

Step 5. Solve the equation. |
[latex]d=42[/latex] miles | |

Step 6. Check |
Does 42 miles make sense? | |

Jamal rides: | ||

Step 7. Answer the question with a complete sentence. |
Jamal rode 42 miles. |

### **Try It #1**

Lindsay drove for [latex]5\frac{1}{2}[/latex] hours at 60 miles per hour. How much distance did she travel?

### **Try It #2**

Trinh walked for [latex]2\frac{1}{3}[/latex] hours at 3 miles per hour. How far did she walk?

**EXAMPLE 2**

**Creating a Mini-Chart to Summarize the Information in the Problem**

Rey is planning to drive from his house in Saskatoon to visit his grandmother in Winnipeg, a distance of 520 miles. If he can drive at a steady rate of 65 miles per hour, how many hours will the trip take?

## Show/Hide Solution

#### Solution

Step 1. Read the problem. |
||

Step 2. Identify what you are looking for. |
How many hours (time) | |

Step 3. Name: Choose a variable to represent it. |
Let [latex]t[/latex] = time. | |

Step 4. Translate: Write the appropriate formula. |
[latex]\phantom{\rule{1em}{0ex}}d=rt[/latex] | |

Substitute in the given information. | [latex]520=65t[/latex] | |

Step 5. Solve the equation. |
[latex]\phantom{\rule{1.2em}{0ex}}t=8[/latex] | |

Step 6. Check. Substitute the numbers intothe formula and make sure the result is a true statement. |
[latex]\begin{array}{ccc}\hfill d& =\hfill & rt\hfill \\ \hfill 520& \stackrel{?}{=}\hfill & 65\cdot 8\hfill \\ \hfill 520& =\hfill & 520\hfill \end{array}[/latex] | |

Step 7. Answer the question with a complete sentence. Rey’s trip will take 8 hours. |

### **Try It #3**

Lee wants to drive from Kamloops to his brother’s apartment in Banff, a distance of 495 km. If he drives at a steady rate of 90 km/h, how many hours will the trip take? Yesenia is 168 km from Toronto. If she needs to be in Toronto in 2 hours, at what rate does she need to drive?

### **Try It #4**

Yesenia is 168 km from Toronto. If she needs to be in Toronto in 2 hours, at what rate does she need to drive?

**Solve a Formula for a Specific Variable**

You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be familiar with formulas and be able to manipulate them easily.

In Example 1 and Example 2, we used the formula [latex]d=rt[/latex]. This formula gives the value of [latex]d[/latex], distance, when you substitute in the values of [latex]\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t[/latex], the rate and time. But in Example 2, we had to find the value of [latex]t[/latex]. We substituted in values of [latex]\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r[/latex] and then used algebra to solve for [latex]t[/latex]. If you had to do this often, you might wonder why there is not a formula that gives the value of [latex]t[/latex] when you substitute in the values of [latex]\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r[/latex]. We can make a formula like this by solving the formula [latex]d=rt[/latex] for [latex]t[/latex].

To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. To see how to solve a formula for a specific variable, we will start with the distance, rate and time formula.

**EXAMPLE 3**

**Solving for a ****Specified ****Variable**

Solve the formula [latex]d=rt[/latex] for [latex]t[/latex]:

- when [latex]d=520[/latex] and [latex]r=65[/latex].
- in general.

## Show/Hide Solution

#### Solution

We will write the solutions side-by-side to demonstrate that solving a formula in general uses the same steps as when we have numbers to substitute.

a) when [latex]d=520[/latex] and [latex]r=65[/latex] |
b) in general |
||||

Write the formula. | [latex]\phantom{\rule{1em}{0ex}}d=rt[/latex] | Write the formula. | [latex]d=rt[/latex] | ||

Substitute. | [latex]520=65t[/latex] | ||||

Divide, to isolate [latex]t[/latex]. | [latex]\frac{520}{65}=\frac{65t}{65}[/latex] | Divide, to isolate [latex]t[/latex]. | [latex]\frac{d}{r}=\frac{rt}{r}[/latex] | ||

Simplify. | [latex]\phantom{\rule{1.2em}{0ex}}8=t[/latex] | Simplify. | [latex]\frac{d}{r}=t[/latex] |

We say the formula [latex]t=\frac{d}{r}[/latex] is solved for [latex]t[/latex].

### **Try It #5**

Solve the formula [latex]d=rt[/latex] for [latex]r[/latex]:

a) when [latex]d=180\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=4[/latex].

b) in general.

### **Try It #6**

Solve the formula [latex]d=rt[/latex] for [latex]r[/latex]:

a) when [latex]d=780\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=12[/latex].

b) in general.

**EXAMPLE 4**

**Solving for a ****Specified ****Variable**

Solve the formula [latex]A=\frac{1}{2}bh[/latex] for [latex]h[/latex]:

a) when [latex]A=90[/latex] and [latex]b=15[/latex].

b) in general.

## Show/Hide Solution

#### Solution

a) when [latex]A=90[/latex] and [latex]b=15[/latex] |
b) in general |
||||

Write the formula. | Write the formula. | ||||

Substitute. | |||||

Clear the fractions. | Clear the fractions. | ||||

Simplify. | Simplify. | ||||

Solve for [latex]h[/latex]. | Solve for [latex]h[/latex]. |

We can now find the height of a triangle, if we know the area and the base, by using the formula [latex]h=\frac{2A}{b}[/latex].

### **Try It #7**

Use the formula [latex]A=\frac{1}{2}bh[/latex] to solve for [latex]h[/latex]:

a) when [latex]A=170[/latex] and [latex]b=17[/latex].

b) in general.

### **Try It #8**

Use the formula [latex]A=\frac{1}{2}bh[/latex] to solve for [latex]b[/latex]:

a) when [latex]A=62[/latex] and [latex]h=31[/latex].

b) in general.

**EXAMPLE 5**

**Solving for a ****Specified ****Variable**

Solve the formula [latex]I=Prt[/latex] to find the principal, [latex]P[/latex]:

a) when [latex]I=\$5,600[/latex], [latex]r=4\%[/latex], [latex]t=7\phantom{\rule{0.2em}{0ex}}years\phantom{\rule{0.2em}{0ex}}[/latex].

b) in general.

## Show/Hide Solution

#### Solution

a) [latex]I=\$5,600[/latex], [latex]r=4\%[/latex], [latex]t=7 years[/latex] |
b) in general |
||

Write the formula. | Write the formula. | ||

Substitute. | |||

Simplify. | Simplify. | ||

Divide, to isolate [latex]P[/latex]. | Divide, to isolate [latex]P[/latex]. | ||

Simplify. | Simplify. | ||

The principal is |

### **Try It #9**

Use the formula [latex]I=Prt[/latex] to find the principal, [latex]P[/latex]:

a) when [latex]I=\$2,160[/latex], [latex]r=6\%[/latex], [latex]t=3\phantom{\rule{0.2em}{0ex}}years\phantom{\rule{0.2em}{0ex}}[/latex].

b) in general.

### **Try It #10**

Use the formula [latex]I=Prt[/latex] to find the principal, [latex]P[/latex]:

a) when [latex]I=\$5,400[/latex], [latex]r=12\%[/latex], [latex]t=5\phantom{\rule{0.2em}{0ex}}years\phantom{\rule{0.2em}{0ex}}[/latex].

b) in general.

**EXAMPLE 6**

**Solving for a ****Specified ****Variable**

Solve the formula [latex]3x+2y=18[/latex] for [latex]y[/latex]:

a) when [latex]x=4[/latex].

b) in general.

## Show/Hide Solution

#### Solution

a) when [latex]x=4[/latex] |
b) in general |
||

Substitute. | |||

Subtract to isolate the [latex]y-[/latex]term. | Subtract to isolate the [latex]y-[/latex]term. | ||

Divide. | Divide. | ||

Simplify. | Simplify. |

### **Try It #11**

Solve the formula [latex]3x+4y=10[/latex] for [latex]y[/latex]:

a) when [latex]x=\frac{14}{3}[/latex].

b) in general.

### **Try It #12**

Solve the formula [latex]5x+2y=18[/latex] for [latex]y[/latex]*:*

a) when [latex]x=4[/latex].

b) in general.

**EXAMPLE 7**

**Solving for a ****Specified ****Variable**

Solve the formula [latex]P=a+b+c[/latex] for [latex]a[/latex].

## Show/Hide Solution

#### Solution

We will isolate [latex]a[/latex] on one side of the equation. | |

Both [latex]b[/latex] and [latex]c[/latex] are added to [latex]a[/latex], so we subtract them from both sides of the equation. | |

Simplify. |

### **Try It #13**

Solve the formula [latex]P=a+b+c[/latex] for [latex]b[/latex].

### **Try It #14**

Solve the formula [latex]P=a+b+c[/latex] for [latex]c[/latex].

**EXAMPLE 8**

**Solving for a Specified Variable**

Solve the formula [latex]6x+5y=13[/latex] for [latex]y[/latex]*.*

## Show/Hide Solution

#### Solution

Subtract [latex]6x[/latex] from both sides to isolate the term with [latex]y[/latex]. | |

Simplify. | |

Divide by 5 to make the coefficient 1. | |

Simplify. |

The fraction is simplified. We cannot divide [latex]13-6x[/latex] by 5.

### **Try It #15**

Solve the formula [latex]4x+7y=9[/latex] for [latex]y[/latex]*.*

### **Try It #16**

Solve the formula [latex]5x+8y=1[/latex] for [latex]y[/latex]*.*

# Section 2.6 Exercises

**[Answers to odd problem numbers are provided at the end of the problem set. Just scroll down!]**

**Use the Distance, Rate, and Time Formula**

In the following exercises, solve for the specified variable.

1. Socorro drove for [latex]4\frac{5}{6}[/latex] hours at 60 miles per hour. How much distance did she travel?

2. Steve drove for [latex]8\frac{1}{2}[/latex] hours at 72 miles per hour. How much distance did he travel?

3. Francie rode her bike for [latex]2\frac{1}{2}[/latex] hours at 12 miles per hour. How far did she ride?

4. Yuki walked for [latex]1\frac{3}{4}[/latex] hours at 4 miles per hour. How far did she walk?

5. Marta is taking the bus from Abbotsford to Cranbrook. The distance is 774 km and the bus travels at a steady rate of 86 miles per hour. How long will the bus ride be?

6. Connor wants to drive from Vancouver to the Nakusp, a distance of 630 km. If he drives at a steady rate of 90 km/h, how many hours will the trip take?

7. Kareem wants to ride his bike from Golden, BC to Banff, AB. The distance is 140 km. If he rides at a steady rate of 20 km/h, how many hours will the trip take?

8. Aurelia is driving from Calgary to Edmonton at a rate of 85 km/h. The distance is 300 km. To the nearest tenth of an hour, how long will the trip take?

9. Alejandra is driving to Prince George, 450 km away. If she wants to be there in 6 hours, at what rate does she need to drive?

10. Javier is driving to Vernon, 240 km away. If he needs to be in Vernon in 3 hours, at what rate does he need to drive?

11. Philip got a ride with a friend from Calgary to Kelowna, a distance of 890 km. If the trip took 10 hours, how fast was the friend driving?

12. Aisha took the train from Spokane to Seattle. The distance is 280 miles and the trip took 3.5 hours. What was the speed of the train?

**Solve a Formula for a Specific Variable**

In the following exercises, use the formula [latex]d=rt[/latex].

13. Solve for [latex]t[/latex]

a) when [latex]d=240\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=60[/latex].

b) in general.

14. Solve for [latex]t[/latex]

a) when [latex]d=350[/latex] and [latex]r=70[/latex].

b) in general.

15. Solve for [latex]t[/latex]

a) when [latex]d=175\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=50[/latex].

b) in general.

16. Solve for [latex]t[/latex]

a) when [latex]d=510\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=60[/latex].

b) in general.

17. Solve for [latex]r[/latex]

a) when [latex]d=420\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=6[/latex].

b) in general.

18. Solve for [latex]r[/latex]

a) when [latex]d=204\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=3[/latex].

b) in general.

19. Solve for [latex]r[/latex]

a) when [latex]d=180\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=4.5[/latex].

b) in general.

20. Solve for [latex]r[/latex]

a) when [latex]d=160\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=2.5[/latex].

b) in general.

In the following exercises, use the formula [latex]A=\frac{1}{2}bh[/latex].

21. Solve for [latex]h[/latex]

a) when [latex]A=176\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=22[/latex].

b) in general.

22. Solve for [latex]b[/latex]

a) when [latex]A=126\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}h=18[/latex].

b) in general.

23. Solve for the principal, P for

a) [latex]I=\$5,480,\,r=4\%,\,t=7\phantom{\rule{0.2em}{0ex}}\text{years}\phantom{\rule{0.2em}{0ex}}[/latex].

b) in general.

24. Solve for [latex]b[/latex]

a) when [latex]A=65\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}h=13[/latex].

b) in general.

In the following exercises, use the formula [latex]I = Prt[/latex].

25. Solve for the time, [latex]t[/latex] for

a) [latex]I=\$2,376,\, P=\$9,000,\,r=4.4\%[/latex].

b) in general.

26. Solve for the principal, P for

a) [latex]I=\$3,950,\,r=6\%,\,t=5\phantom{\rule{0.2em}{0ex}}\text{years}\phantom{\rule{0.2em}{0ex}}[/latex].

b) in general.

27. Solve the formula [latex]2x+3y=12[/latex] for [latex]y[/latex]

a) when [latex]x=3[/latex].

b) in general.

28. Solve for the time, [latex]t[/latex] for

a) [latex]I=\$624,\,P=\$6,000,\,r=5.2\%[/latex].

b) in general.

In the following exercises, solve.

29. Solve the formula [latex]3x-y=7[/latex] for [latex]y[/latex]

a) when [latex]x=-2[/latex].

b) in general.

30. Solve the formula [latex]5x+2y=10[/latex] for [latex]y[/latex]

a) when [latex]x=4[/latex].

b) in general.

31. Solve [latex]a+b=90[/latex] for [latex]b[/latex].

32. Solve the formula [latex]4x+y=5[/latex] for [latex]y[/latex]

a) when [latex]x=-3[/latex].

b) in general.

33. Solve [latex]180=a+b+c[/latex] for [latex]a[/latex].

34. Solve [latex]a+b=90[/latex] for [latex]a[/latex].

35. Solve the formula [latex]8x+y=15[/latex] for [latex]y[/latex].

36. Solve [latex]180=a+b+c[/latex] for [latex]c[/latex].

37. Solve the formula [latex]-4x+y=-6[/latex] for [latex]y[/latex].

38. Solve the formula [latex]9x+y=13[/latex] for [latex]y[/latex].

39. Solve the formula [latex]4x+3y=7[/latex] for [latex]y[/latex].

40. Solve the formula [latex]-5x+y=-1[/latex] for [latex]y[/latex].

41. Solve the formula [latex]x-y=-4[/latex] for [latex]y[/latex].

42. Solve the formula [latex]3x+2y=11[/latex] for [latex]y[/latex].

43. Solve the formula [latex]P=2L+2W[/latex] for [latex]L[/latex].

44. Solve the formula [latex]x-y=-3[/latex] for [latex]y[/latex].

45. Solve the formula [latex]C=\pi d[/latex] for [latex]d[/latex].

46. Solve the formula [latex]P=2L+2W[/latex] for [latex]W[/latex].

47. Solve the formula [latex]V=LWH[/latex] for [latex]L[/latex].

48. Solve the formula [latex]C=\pi d[/latex] for [latex]\pi[/latex].

49. Solve the formula [latex]V=LWH[/latex] for[latex]H[/latex].

**Everyday Math**

50. Converting temperature. Yon was visiting the United States and he saw that the temperature in Seattle one day was 50o Fahrenheit. Solve for C in the formula [latex]F=\frac{9}{5}C+32[/latex] to find the Celsius temperature.

51. Converting temperature. While on a tour in Greece, Tatyana saw that the temperature was 40o Celsius. Solve for F in the formula [latex]C=\frac{5}{9}\left(F-32\right)[/latex] to find the Fahrenheit temperature.

**Writing Exercises**

52. Solve the equation [latex]5x-2y=10[/latex] for [latex]x[/latex]

a) when [latex]y=10[/latex].

b) in general.

c) Which solution is easier for you, a) or b)? Why?

53. Solve the equation [latex]2x+3y=6[/latex] for [latex]y[/latex]

a) when [latex]x=-3[/latex].

b) in general.

c) Which solution is easier for you, a) or b)? Why?

# Answers to Section 2.6 Odd Problems

1. 290 miles

3. 30 miles

5. 9 hours

7. 75 km/h

9. 3.5 hours

11. 7 hours

13. 7

15. 89 km/h

17. a) [latex]t=4[/latex]

b) [latex]t=\frac{d}{r}[/latex]

19. a) [latex]t=3.5[/latex]

b) [latex]t=\frac{d}{r}[/latex]

21. a) [latex]r=70[/latex]

b) [latex]r=\frac{d}{t}[/latex]

23. a) [latex]r=40[/latex]

b) [latex]r=\frac{d}{t}[/latex]

25. a) [latex]h=16[/latex]

b) [latex]h=\frac{2A}{b}[/latex]

27. a) [latex]b=10[/latex]

b) [latex]b=\frac{2A}{h}[/latex]

29. a) [latex]P=\$13,166.67[/latex]

b) [latex]P=\frac{I}{rt}[/latex]

31. a) [latex]t=2[/latex] years

b) [latex]t=\frac{I}{\mathrm{Pr}}[/latex]

33. a) [latex]y=-5[/latex]

b) [latex]y=\frac{10-5x}{2}[/latex]

35. a) [latex]y=17[/latex]

b) [latex]y=5-4x[/latex]

37. [latex]a=90-b[/latex]

39. [latex]c=180-a-b[/latex]

41. [latex]y=13-9x[/latex]

43. [latex]y=-1+5x[/latex]

45. [latex]y=\frac{11-3x}{4}[/latex]

47. [latex]y=3+x[/latex]

49. [latex]W=\frac{P-2L}{2}[/latex]