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Solve a Formula for a Specific Variable

Use the distance, rate, and time formula

Use the Distance, Rate, and Time Formula

One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of \(60\) miles per hour for \(2\) hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said \(120\) miles, you already know how to use this formula!

The math to calculate the distance might look like this:

\[\begin{array}{l} \\ \text{distance}=(\frac{60\ \text{miles}}{1\ \text{hour}})(2\ \text{hours}) \\ \text{distance}=120\ \text{miles}\end{array}\]

In general, the formula relating distance, rate, and time is

\[\text{distance}\ \text{=}\ \text{rate}\cdot \text{time}\]

Notice that the units we used above for the rate were miles per hour, which we can write as a ratio \(\frac{miles}{hour}.\) Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.

Example

Try it.

Jamal rides his bike at a uniform rate of \(12\) miles per hour for \(3\frac{1}{2}\) hours. How much distance has he traveled?

Solution
Step 1. Read the problem.
You may want to create a mini-chart to summarize the
information in the problem.		\(d=?\)
\(r=12\ \text{mph}\)
\(t=3\frac{1}{2}\ \text{hours}\)
Step 2. Identify what you are looking for.distance traveled
Step 3. Name. Choose a variable to represent it.let d = distance
Step 4. Translate.
Write the appropriate formula for the situation.
Substitute in the given information.		\(d=rt\)

\(d=12⋅3\frac{1}{2}\)
Step 5. Solve the equation.\(d=42\ \text{miles}\)
Step 6. Check: Does 42 miles make sense?
Step 7. Answer the question with a complete sentence.Jamal rode 42 miles.

Condensed — the full section is in OpenStax Prealgebra 2e.

Solve a Formula for a Specific Variable

In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft ExcelTM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.

In and , we used the formula \(d=rt.\) This formula gives the value of \(d\) when you substitute in the values of \(r\) and \(t.\) But in , we had to find the value of \(t.\) We substituted in values of \(d\) and \(r\) and then used algebra to solve to \(t.\) If you had to do this often, you might wonder why there isn’t a formula that gives the value of \(t\) when you substitute in the values of \(d\) and \(r.\) We can get a formula like this by solving the formula \(d=rt\) for \(t.\)

To solve a formula for a specific variable means to get that variable by itself with a coefficient of \(1\) on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.

Let’s try a few examples, starting with the distance, rate, and time formula we used above.

Example

Try it.

Solve the formula \(d=rt\) for \(t\text{:}\)

  1. ⓐ when \(d=520\) and \(r=65\)
  2. ⓑ in general.

Solution

We’ll write the solutions side-by-side so you can see 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 forumla.
Substitute any given values.
Divide to isolate t.
Simplify.

Notice that the solution for ⓐ is the same as that in . We say the formula \(t=\frac{d}{r}\) is solved for \(t.\) We can use this version of the formula anytime we are given the distance and rate and need to find the time.

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Write \(35\) miles per gallon as a unit rate.
    If you missed this problem, review .

    เปิดเผยคำตอบ

    \(\frac{35\ \text{miles}}{1\ \text{gallon}}\)

  2. Solve \(6x+24=96.\)
    If you missed this problem, review .

    เปิดเผยคำตอบ

    \(12\)

  3. Find the simple interest earned after \(5\) years on \(\text{\$1,000}\) at an interest rate of \(\text{4\%}.\)
    If you missed this problem, review .

    เปิดเผยคำตอบ

    \(\$200\)

  4. Jamal rides his bike at a uniform rate of \(12\) miles per hour for \(3\frac{1}{2}\) hours. How much distance has he traveled?

    เปิดเผยคำตอบ
    Step 1. Read the problem.
    You may want to create a mini-chart to summarize the
    information in the problem.    		\(d=?\)
    \(r=12\ \text{mph}\)
    \(t=3\frac{1}{2}\ \text{hours}\)
    Step 2. Identify what you are looking for.distance traveled
    Step 3. Name. Choose a variable to represent it.let d = distance
    Step 4. Translate.
    Write the appropriate formula for the situation.
    Substitute in the given information.    		\(d=rt\)

    \(d=12⋅3\frac{1}{2}\)
    Step 5. Solve the equation.\(d=42\ \text{miles}\)
    Step 6. Check: Does 42 miles make sense?
    Step 7. Answer the question with a complete sentence.Jamal rode 42 miles.

  5. Lindsay drove for \(5\frac{1}{2}\) hours at \(60\) miles per hour. How much distance did she travel?

    เปิดเผยคำตอบ

    330 mi

  6. Trinh walked for \(2\frac{1}{3}\) hours at \(3\) miles per hour. How far did she walk?

    เปิดเผยคำตอบ

    7 mi

  7. Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of \(520\) miles. If he can drive at a steady rate of \(65\) miles per hour, how many hours will the trip take?

    เปิดเผยคำตอบ
    Step 1. Read the problem.
    Summarize the information in the problem.    		\(d=520\ \text{miles}\)
    \(r=65\ \text{mph}\)
    \(t=?\)
    Step 2. Identify what you are looking for.how many hours (time)
    Step 3. Name:
    Choose a variable to represent it.    		let t = time
    Step 4. Translate.
    Write the appropriate formula.
    Substitute in the given information.    		\(d=rt\)
    \(520=65t\)
    Step 5. Solve the equation.\(t=8\)
    Step 6. Check:
    Substitute the numbers into the formula and make sure
    the result is a true statement.
    \(d=rt\)
    \(520\overset{?}{=}65⋅8\)
    \(520=520>✓\)
    Step 7. Answer the question with a complete sentence.
    We know the units of time will be hours because
    we divided miles by miles per hour.    		Rey's trip will take 8 hours.

  8. 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?

    เปิดเผยคำตอบ

    11 hours

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

    เปิดเผยคำตอบ

    56 mph

  10. Solve the formula \(d=rt\) for \(t\text{:}\)

    1. ⓐ when \(d=520\) and \(r=65\)
    2. ⓑ in general.

    เปิดเผยคำตอบ

    We’ll write the solutions side-by-side so you can see 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 forumla.
    Substitute any given values.
    Divide to isolate t.
    Simplify.

    Notice that the solution for ⓐ is the same as that in . We say the formula \(t=\frac{d}{r}\) is solved for \(t.\) We can use this version of the formula anytime we are given the distance and rate and need to find the time.

  11. Solve the formula \(d=rt\) for \(r\text{:}\)

    1. ⓐ when \(d=180\) and \(t=4\)
    2. ⓐ in general

    เปิดเผยคำตอบ

    1. ⓐ \(\ r=45\)
    2. ⓑ \(\ r=\frac{d}{t}\)

  12. Solve the formula \(d=rt\) for \(r\text{:}\)

    1. ⓐ when \(d=780\) and \(t=12\)
    2. ⓑ in general

    เปิดเผยคำตอบ
    1. ⓐ \(\ r=65\)
    2. ⓑ \(\ r=\frac{d}{t}\)

  13. The formula for area of a triangle is \(A=\frac{1}{2}bh.\) Solve this formula for \(h\text{:}\)

    1. ⓐ when \(A=90\) and \(b=15\)
    2. ⓑ in general

    เปิดเผยคำตอบ
    ⓐ when A = 90 and b = 15ⓑ in general
    Write the forumla.
    Substitute any given values.
    Clear the fractions.
    Simplify.
    Solve for h.

    We can now find the height of a triangle, if we know the area and the base, by using the formula

    \[h=\frac{2A}{b}\]

  14. Use the formula \(A=\frac{1}{2}bh\) to solve for \(h:\)

    1. ⓐ when \(A=170\) and \(b=17\)
    2. ⓑ in general

    เปิดเผยคำตอบ

    ⓐ \(\ h=20\) ⓑ \(\ h=\frac{2A}{b}\)

  15. Use the formula \(A=\frac{1}{2}bh\) to solve for \(b\text{:}\)

    1. ⓐ when \(A=62\) and \(h=31\)
    2. ⓑ in general

    เปิดเผยคำตอบ
    1. ⓐ \(\ b=4\)
    2. ⓑ \(\ b=\frac{2A}{h}\)

  16. Solve the formula \(I=Prt\) to find the principal, \(P\text{:}\)

    1. ⓐ when \(I=\text{\$5,600},\ r=\text{4\%},\ t=7\ \text{years}\)
    2. ⓑ in general

    เปิดเผยคำตอบ
    I = $5600, r = 4%, t = 7 yearsin general
    Write the forumla.
    Substitute any given values.
    Multiply r ⋅ t.
    Divide to isolate P.
    Simplify.
    State the answer.The principal is $20,000.

  17. Use the formula \(I=Prt.\)

    Find \(t\text{:}\) ⓐ when \(I=\text{\$2,160},\ r=\text{6\%},\ P=\text{\$12,000;}\) ⓑ in general

    เปิดเผยคำตอบ

    1. ⓐ \(\ t=\text{3 years}\\)
    2. ⓑ \(\ t=\frac{I}{P\ r}\)

  18. Use the formula \(I=Prt.\)

    Find \(r\text{:}\) ⓐ when \(I=\text{\$5,400},\ P=\text{\$9,000},\ t=5\ \text{years}\) ⓑ in general

    เปิดเผยคำตอบ

    1. ⓐ \(\ r=0.12=\ \text{12\%}\\)
    2. ⓑ \(\ r=\frac{I}{P\ t}\)

  19. Solve the formula \(3x+2y=18\) for \(y\text{:}\)

    1. ⓐ when \(x=4\)
    2. ⓑ in general

    เปิดเผยคำตอบ
    when x = 4in general
    Write the equation.
    Substitute any given values.
    Simplify if possible.
    Subtract to isolate the y-term.
    Simplify.
    Divide.
    Simplify.

  20. Solve the formula \(3x+4y=10\) for \(y\text{:}\)

    1. ⓐ when \(x=2\)
    2. ⓑ in general

    เปิดเผยคำตอบ

    1. ⓐ \(\ y=1\)
    2. ⓑ \(\ y=\frac{10-3x}{4}\)

  21. Solve the formula \(5x+2y=18\) for \(y\text{:}\)

    1. ⓐ when \(x=4\)
    2. ⓑ in general

    เปิดเผยคำตอบ

    1. ⓐ \(\ y=-1\)
    2. ⓑ \(\ y=\frac{18-5x}{2}\)

  22. Solve the formula \(P=a+b+c\) for \(a.\)

    เปิดเผยคำตอบ

    We will isolate \(a\) on one side of the equation.

    We will isolate a on one side of the equation.
    Write the equation.\(P=a+b+c\)
    Subtract b and c from both sides to isolate a.
    Simplify.\(P-b-c=a\)

    So, \(a=P-b-c\)

  23. Solve the formula \(P=a+b+c\) for \(b\text{.}\)

    เปิดเผยคำตอบ

    b = P − a − c

  24. Solve the formula \(P=a+b+c\) for \(c\text{.}\)

    เปิดเผยคำตอบ

    c = P − a − b

  25. Solve the equation \(3x+y=10\) for \(y.\)

    เปิดเผยคำตอบ

    We will isolate \(y\) on one side of the equation.

    We will isolate y on one side of the equation.
    Write the equation.\(3x+y=10\)
    Subtract 3x from both sides to isolate y.
    Simplify.\(y=10-3x\)

  26. Solve the formula \(7x+y=11\) for \(y.\)

    เปิดเผยคำตอบ

    y = 11 − 7x

  27. Solve the formula \(11x+y=8\) for \(y.\)

    เปิดเผยคำตอบ

    y = 8 − 11x

  28. Solve the equation \(6x+5y=13\) for \(y.\)

    เปิดเผยคำตอบ

    We will isolate \(y\) on one side of the equation.

    We will isolate y on one side of the equation.
    Write the equation.
    Subtract to isolate the term with y.
    Simplify.
    Divide 5 to make the coefficient 1.
    Simplify.

  29. Solve the formula \(4x+7y=9\) for \(y.\)

    เปิดเผยคำตอบ

    \(y=\frac{9-4x}{7}\)

  30. Solve the formula \(5x+8y=1\) for \(y.\)

    เปิดเผยคำตอบ

    \(y=\frac{1-5x}{8}\)

  31. Steve drove for \(8\frac{1}{2}\) hours at \(72\) miles per hour. How much distance did he travel?

    เปิดเผยคำตอบ

    612 mi

  32. Socorro drove for \(4\frac{5}{6}\) hours at \(60\) miles per hour. How much distance did she travel?

  33. Yuki walked for \(1\frac{3}{4}\) hours at \(4\) miles per hour. How far did she walk?

    เปิดเผยคำตอบ

    7 mi

  34. Francie rode her bike for \(2\frac{1}{2}\) hours at \(12\) miles per hour. How far did she ride?

  35. Connor wants to drive from Tucson to the Grand Canyon, a distance of \(338\) miles. If he drives at a steady rate of \(52\) miles per hour, how many hours will the trip take?

    เปิดเผยคำตอบ

    6.5 hours

  36. Megan is taking the bus from New York City to Montreal. The distance is \(384\) miles and the bus travels at a steady rate of \(64\) miles per hour. How long will the bus ride be?

  37. Aurelia is driving from Miami to Orlando at a rate of \(65\) miles per hour. The distance is \(235\) miles. To the nearest tenth of an hour, how long will the trip take?

    เปิดเผยคำตอบ

    3.6 hours

  38. Kareem wants to ride his bike from St. Louis, Missouri to Champaign, Illinois. The distance is \(180\) miles. If he rides at a steady rate of \(16\) miles per hour, how many hours will the trip take?

  39. Javier is driving to Bangor, Maine, which is \(240\) miles away from his current location. If he needs to be in Bangor in \(4\) hours, at what rate does he need to drive?

    เปิดเผยคำตอบ

    60 mph

  40. Alejandra is driving to Cincinnati, Ohio, \(450\) miles away. If she wants to be there in \(6\) hours, at what rate does she need to drive?

Symbols used here

f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\approx
approximately equal
Equal to the precision shown, not exactly.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
a \bmod n
remainder
What is left after dividing a by n.
\%
per cent
Per hundred: 15% = 15/100.
a : b,\ \frac{a}{b}
ratio, fraction
a for every b; a divided by b.

How to: Solve a Formula for a Specific Variable

  1. Use the distance, rate, and time formula
  2. Solve a formula for a specific variable

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

ลองดูสิ

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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