# 2.6 Solve a formula for a specific variable  (Page 2/4)

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Lee wants to drive from Phoenix to his brother’s apartment in San Francisco, a distance of 770 miles. If he drives at a steady rate of 70 miles per hour, how many hours will the trip take?

11 hours

Yesenia is 168 miles from Chicago. If she needs to be in Chicago in 3 hours, at what rate does she need to drive?

56 mph

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

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.

Solve the formula $d=rt$ for $t$ :

1. when $d=520$ and $r=65$
2. in general

## 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.

 ⓐ when $d=520$ and $r=65$ ⓑ in general Write the formula. $\phantom{\rule{1em}{0ex}}d=rt$ Write the formula. $d=rt$ Substitute. $520=65t$ Divide, to isolate $t$ . $\frac{520}{65}=\frac{65t}{65}$ Divide, to isolate $t$ . $\frac{d}{r}=\frac{rt}{r}$ Simplify. $\phantom{\rule{1.2em}{0ex}}8=t$ Simplify. $\frac{d}{r}=t$

We say the formula $t=\frac{d}{r}$ is solved for $t$ .

Solve the formula $d=rt$ for $r$ :

when $d=180\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=4$ in general

$r=45$ $r=\frac{d}{t}$

Solve the formula $d=rt$ for $r$ :

when $d=780\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=12$ in general

$r=65$ $r=\frac{d}{t}$

Solve the formula $A=\frac{1}{2}bh$ for $h$ :

when $A=90$ and $b=15$ in general

## Solution

 ⓐ when $A=90$ and $b=15$ ⓑ in general Write the formula. Write the formula. Substitute. Clear the fractions. Clear the fractions. Simplify. Simplify. Solve for $h$ . Solve for $h$ .

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

Use the formula $A=\frac{1}{2}bh$ to solve for $h$ :

when $A=170$ and $b=17$ in general

$h=20$ $h=\frac{2A}{b}$

Use the formula $A=\frac{1}{2}bh$ to solve for $b$ :

when $A=62$ and $h=31$ in general

$b=4$ $b=\frac{2A}{h}$

The formula $I=Prt$ is used to calculate simple interest, I , for a principal, P , invested at rate, r , for t years.

Solve the formula $I=Prt$ to find the principal, $P$ :

when $I=\text{}5,600,r=4%,t=7\phantom{\rule{0.2em}{0ex}}\text{years}\phantom{\rule{0.2em}{0ex}}$ in general

## Solution

 ⓐ $I=5,600$ , $r=4%$ , $t=7 years$ ⓑ in general Write the formula. Write the formula. Substitute. Simplify. Simplify. Divide, to isolate P . Divide, to isolate P . Simplify. Simplify. The principal is

Use the formula $I=Prt$ to find the principal, $P$ :

when $I=\text{}2,160,r=6%,t=3\phantom{\rule{0.2em}{0ex}}\text{years}\phantom{\rule{0.2em}{0ex}}$ in general

$12,000 $P=\frac{I}{rt}$ #### Questions & Answers Aziza is solving this equation-2(1+x)=4x+10 Sechabe Reply No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product? KAJAL Reply x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places Adedamola Reply please the answer to the example exercise Patricia Reply 3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training? Edwin Reply .473 Angelita ? Angelita John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18. DaYoungan Reply p-2/3=5/6 how do I solve it with explanation pls Adedamola Reply P=3/2 Vanarith 1/2p2-2/3p=5p/6 James don't understand answer Cindy 4.5 Ruth is y=7/5 a solution of 5y+3=10y-4 Adedamola Reply yes James don't understand answer Cindy Lucinda has a pocketful of dimes and quarters with a value of$6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
is that a negative 4 or positive 4?
Felix
y = mx + b
Felix
if negative -4, then -4=8/9(0) + b
Felix
-4=b
Felix
if positive 4, then 4=b
Felix
then plug in y=8/9x - 4 or y=8/9x+4
Felix
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost$5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make? Cherry Reply Nga and Lauren bought a chest at a flea market for$50. They re-finished it and then added a 350 % mark - up
\$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
2, 17
Jose
interesting
saw
4,2
Cindy
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
hola saben como aser un valor de la expresión
NAILEA
integer greater than 2 and less than 12
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy