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

 Page 2 / 4

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 a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions vinaya Reply a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions vinaya a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions vinaya a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions vinaya Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs$6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs$5.71 per gallon?
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
4x-7y=8 2x-7y=1 what is the answer?
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Frank
thanks
Ramil
copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x
What is c+4=8
2
Letha
4
Lolita
4
Rich
4
thinking
C+4=8 -4 -4 C =4
thinking
I need to study
Letha
4+4=8
William
During two years in college, a student earned $9,500. The second year, she earned$500 more than twice the amount she earned the first year.
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp
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.
Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is$3 and for a package the cost is $7. Ida Reply hey Juan Sup patrick The sum of two numbers is 155. The difference is 23. Find the numbers Michelle Reply The sum of two numbers is 155. Their difference is 23. Find the numbers Michelle The difference between 89 and 66 is 23 Ciid Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution? Amber Reply hello bismark I need a math tutor BAD Stacie Me too Letha me too Xavier ok Bishal teet Bishal Sue and Deb work together writing a book that takes them 90 days. If Sue worked alone, it would take her 120 days. How long would it take Deb to write the book alone? Hailey Reply Tell Deb to write the book alone and report back on how long it took her. Deb could get bored and stop, she could get sick next week and prolong the time, she could die next month. This question is impossible to answer. vV right* tyler is there a thumbs up mathematical symbol in algebra? 👍²= vV 90days x 120days =10800 120days - 90days = 30 10800days/30days= 360days? (ad) derall=1440 mins (90x ad )+(120 x ad )/(120 x ad ))-((90d x ad)= you go glen coco! I'll see my self out Nick Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height? genevieve Reply 6.25 Ciid 6.25 Big ? Twila Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio. Katherine Reply complexed. could you please help me figure out? Ciid (sin=opp/adj) (tan= opp/adj) cos=hyp/adj tyler (sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up tyler (sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up tyler (sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up tyler SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent. tyler H=57 and O=285 figure out what the adjacent? tyler help Twila draw a diagram first John Amara currently sells televisions for company A at a salary of$17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of$29,000 plus a \$20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid