Try It #1

 Try It #2

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 Try It #3

 Try It #4

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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.

 Try It #5

 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.

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 Try It #7

 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.

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 Try It #9

 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.

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 Try It #11

 Try It #12

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

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

b) in general.

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 Try It #13

 Try It #14

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

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 Try It #15

 Try It #16

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

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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?

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.

 

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.

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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]