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 He charges$125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least$2,400?
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
what is algebra
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
rectangular field solutions
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was$1,299. To the nearest tenth of a percent, what was the rate of discount?
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from$1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2% Orlando you could also just divide$725/$1299 and then subtract it from 1. then you get the same answer. Orlando p mulripied-5 and add 30 to it Tausif Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif -5p+30? Corey p=-5+30 Jacob How do you find divisible numbers without a calculator? Jacob Reply TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13 BAINAMA When she graduates college, Linda will owe$43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan? Ariana Reply Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus? Kirisma Reply 66miles/hour snigdha How did you work it out? Esther s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr Orlando hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! Heather look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number... Felix for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer... Felix —12 Niazmohammad Thanks Felix.l also get confused with signs. Esther Thank you for this Shatey ty Graham think about it like you lost$19 (-19), then found $7(+7). Totally you lost just$12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed