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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

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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

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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 [link] and [link] , we used the formula d = r t . This formula gives the value of d , distance, when you substitute in the values of r and t , the rate and time. But in [link] , we had to find the value of t . We substituted in values of d and 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 and r . We can make a formula like this by solving the formula d = r t 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 = r t 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. d = r t Write the formula. d = r t
Substitute. 520 = 65 t
Divide, to isolate t . 520 65 = 65 t 65 Divide, to isolate t . d r = r t r
Simplify. 8 = t Simplify. d r = t

We say the formula t = d r is solved for t .

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Solve the formula d = r t for r :

when d = 180 and t = 4 in general

r = 45 r = d t

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Solve the formula d = r t for r :

when d = 780 and t = 12 in general

r = 65 r = d t

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Solve the formula A = 1 2 b h 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 = 2 A b .

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Use the formula A = 1 2 b h to solve for h :

when A = 170 and b = 17 in general

h = 20 h = 2 A b

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Use the formula A = 1 2 b h to solve for b :

when A = 62 and h = 31 in general

b = 4 b = 2 A h

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

Solve the formula I = P r t to find the principal, P :

when I = $ 5,600 , r = 4 % , t = 7 years 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 . .
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Use the formula I = P r t to find the principal, P :

when I = $ 2,160 , r = 6 % , t = 3 years in general

$12,000 P = I r t

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Questions & Answers

how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply
What is observation
adeyemi Reply
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
Sunnyshay Reply
5 times max
Anton
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 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Sophia Reply
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs $9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot?
Parker Reply
what you wanna get
Cesar
800 sq. ft @ $1.50 & 200 sq. ft @ $9.00
Marco
Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
Enith Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Alexandra Reply
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Cecilia Reply
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year
Heidi Reply
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Elbert Reply
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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