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Mathematics LibreTexts

2.6: Solve a Formula for a Specific Variable

  • Page ID
    30370
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    Skills to Develop

    By the end of this section, you will be able to:

    • Use the Distance, Rate, and Time formula
    • Solve a formula for a specific variable

    Note

    Before you get started, take this readiness quiz.

    1. Solve: 15t=120.
      If you missed this problem, review [link].
    2. Solve: 6x+24=96.
      If you missed this problem, review [link].

    Use the Distance, Rate, and Time Formula

    One formula you will use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant rate. Rate is an equivalent word for “speed.” The basic idea of rate may already familiar to you. Do you know what distance you travel if you drive 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 highway.) If you said 120 miles, you already know how to use this formula!

    DISTANCE, RATE, AND TIME

    For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula:

    \[\begin{array} {lllll}{ d = r t} &{\text { where }} &{ d} &{=} &{\text{distance}} \\ {} &{} &{ r} &{=} &{\text{rate}} \\{} &{} &{ t} &{=} &{\text{time}} \end{array}\]

    We will use the Strategy for Solving Applications that we used earlier in this chapter. When our problem requires a formula, we change Step 4. In place of writing a sentence, we write the appropriate formula. We write the revised steps here for reference.

    SOLVE AN APPLICATION (WITH A FORMULA).

    1. Read the problem. Make sure all the words and ideas are understood.
    2. Identify what we are looking for.
    3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information.
    5. Solve the equation using good algebra techniques.
    6. Check the answer in the problem and make sure it makes sense.
    7. Answer the question with a complete sentence.

    You may want to create a mini-chart to summarize the information in the problem. See the chart in this first example.

    Exercise \(\PageIndex{1}\)

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

    Answer
    Step 1. Read the problem.
    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. \(d=rt\)
    .
    Substitute in the given information. \(d = 12\cdot 3\frac{1}{2}\)
    Step 5. Solve the equation. \(d=42\text{ miles}\)
    Step 6. Check
    Does 42 miles make sense?
    Jamal rides:
    .
    Step 7. Answer the question with a complete sentence. Jamal rode 42 miles.

    Exercise \(\PageIndex{2}\)

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

    Answer

    330 miles

    Exercise \(\PageIndex{3}\)

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

    Answer

    7 miles

    Exercise \(\PageIndex{4}\)

    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?

    Answer
    Step 1. Read the problem.
    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.
    \(d=rt\)
    Substitute in the given information. \(520 = 65t\)
    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.
    \(\begin{array}{lll} {d} &{=} &{rt} \\ {520} &{\stackrel{?}{=}} &{65\cdot 8}\\ {520} &{=} &{520\checkmark} \end{array}\)
    Step 7. Answer the question with a complete sentence. Rey’s trip will take 8 hours.

    Exercise \(\PageIndex{5}\)

    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?

    Answer

    11 hours

    Exercise \(\PageIndex{6}\)

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

    Answer

    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 Exercise \(\PageIndex{1}\) and Exercise \(\PageIndex{4}\), we used the formula \(d=rt\). This formula gives the value of d, distance, when you substitute in the values of r and t, the rate and time. But in Exercise \(\PageIndex{4}\), we had to find the value of t. We substituted in values of d and r and then used algebra to solve for tt. 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 \(d=rt\) 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.

    Exercise \(\PageIndex{7}\)

    Solve the formula d=rt for t:

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

    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.

    1. when d=520 and r=65 2. in general
    Write the formula. \(d=rt\) Write the formula. \(d=rt\)
    Substitute. \(520=65t\)
    Divide, to isolate t. \(\frac{520}{65} = \frac{65t}{65}\) Divide, to isolate tt. \(\frac{d}{r} = \frac{rt}{t}\)
    Simplify. \(8 = t\) Simplify. \(\frac{d}{r}=t\)

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

    Exercise \(\PageIndex{8}\)

    Solve the formula \(d=rt\) for r:

    1. when d=180 and t=4
    2. in general
    Answer
    1. \(r = 45\)
    2. \(r = \frac{d}{t}\)

    Exercise \(\PageIndex{9}\)

    Solve the formula \(d=rt\) for r:

    1. when d=780 and t=12
    2. in general
    Answer
    1. \(r = 65\)
    2. \(r = \frac{d}{rt\)

    Exercise \(\PageIndex{10}\)

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

    1. when \(A = 90\) and \(b = 15\)
    2. in general
    Answer
    1. when \(A = 90\) and \(b = 15\) 2. in general
    Write the formula. . Write the formula. .
    Substitute. .
    Clear the fractions. . Clear the fractions. .
    Simplify. . Simplify. .
    Solve for h. . Solve for hh. .
    We can now find the height of a triangle, if we know the area and the base, by using the formula \(h = \frac{2A}{b}\)

    Exercise \(\PageIndex{11}\)

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

    1. when \(A = 170\) and \(b = 17\)
    2. in general
    Answer
    1. \(h = 20\)
    2. \(h = \frac{2A}{b}\)

    Exercise \(\PageIndex{12}\)

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

    1. when \(A = 62\) and \(h = 31\)
    2. in general
    Answer
    1. \(b = 4\)
    2. \(b = \frac{2A}{h}\)

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

    Exercise \(\PageIndex{13}\)

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

    1. when I=$5,600, r=4% ,t=7years
    2. in general
    Answer
    1. I=$5,600, r=4% ,t=7years 2. in general
    Write the formula. . Write the formula. .
    Substitute. .
    Simplify. . Simplify. .
    Divide, to isolate P. . Divide, to isolate P. .
    Simplify. . Simplify. .
    The principal is . .

    Exercise \(\PageIndex{14}\)

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

    1. when I=$2160, r=6% ,t=3 years
    2. in general
    Answer
    1. $12000
    2. \(P = \frac{1}{rt}\)

    Exercise \(\PageIndex{15}\)

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

    1. when I=$5400, r= 12% ,t=5 years
    2. in general
    Answer
    1. $9000
    2. \(P = \frac{1}{rt}\)
    Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually x and y. You might be given an equation that is solved for y and need to solve it for x, or vice versa. In the following example, we’re given an equation with both x and y on the same side and we’ll solve it for y.

    Exercise \(\PageIndex{16}\)

    Solve the formula 3x+2y=18 for y:

    1. when x=4
    2. in general
    Answer
    1. when x=4 2. in general
    . .
    Substitute. .
    Subtract to isolate the
    y-term.
    . Subtract to isolate the
    y-term.
    .
    Divide. . Divide. .
    Simplify. . Simplify. .

    Exercise \(\PageIndex{17}\)

    Solve the formula 3x+4y=10 for y:

    1. when \(x = \frac{14}{3}\)
    2. in general
    Answer
    1. \(y = 1\)
    2. \(y = \frac{10 - 3x}{4}\)

    Exercise \(\PageIndex{18}\)

    Solve the formula 5x+2y=18 for y:

    1. when \(x = 4\)
    2. in general
    Answer
    1. \(y = -1\)
    2. \(y = \frac{18 - 5x}{2}\)
    In Examples 1.60 through 1.64 we used the numbers in part 1 as a guide to solving in general in part 2. Now we will solve a formula in general without using numbers as a guide.

    Exercise \(\PageIndex{19}\)

    Solve the formula P=a+b+c for a.

    Answer
    We will isolate aa on one side of the equation. .
    Both b and c are added to a, so we subtract them from both sides of the equation. .
    Simplify. .
    .

    Exercise \(\PageIndex{20}\)

    Solve the formula P=a+b+c for b.

    Answer

    b=P−a−c

    Exercise \(\PageIndex{21}\)

    Solve the formula P=a+b+c for c.

    Answer

    c=P−a−b

    Exercise \(\PageIndex{22}\)

    Solve the formula 6x+5y=13 for y.

    Answer
    .
    Subtract 6x from both sides to isolate the term with y. .
    Simplify. .
    Divide by 5 to make the coefficient 1. .
    Simplify. .

    The fraction is simplified. We cannot divide 13−6x by 5.

    Exercise \(\PageIndex{23}\)

    Solve the formula 4x+7y=9 for y.

    Answer

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

    Exercise \(\PageIndex{24}\)

    Solve the formula 5x+8y=1 for y.

    Answer

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

    Key Concepts

    • To Solve an Application (with a formula)
      1. Read the problem. Make sure all the words and ideas are understood.
      2. Identify what we are looking for.
      3. Name what we are looking for. Choose a variable to represent that quantity.
      4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information.
      5. Solve the equation using good algebra techniques.
      6. Check the answer in the problem and make sure it makes sense.
      7. Answer the question with a complete sentence.
    • Distance, Rate and Time
      For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula: d=rt where d = distance, r = rate, t = time.
    • 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 other variables and constants on the other side.

    Practice Makes Perfect

    Use the Distance, Rate, and Time Formula

    In the following exercises, solve.

    Exercise \(\PageIndex{1}\)

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

    Exercise \(\PageIndex{2}\)

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

    Answer

    290 miles

    Exercise \(\PageIndex{3}\)

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

    Exercise \(\PageIndex{4}\)

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

    Answer

    30 miles

    Exercise \(\PageIndex{5}\)

    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?

    Exercise \(\PageIndex{6}\)

    Megan is taking the bus from New York City to Montreal. The distance is 380 miles and the bus travels at a steady rate of 76 miles per hour. How long will the bus ride be?

    Answer

    5 hours

    Exercise \(\PageIndex{7}\)

    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?

    Exercise \(\PageIndex{8}\)

    Kareem wants to ride his bike from St. Louis 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?

    Answer

    11.25 hours

    Exercise \(\PageIndex{9}\)

    Javier is driving to Bangor, 240 miles away. If he needs to be in Bangor in 4 hours, at what rate does he need to drive?

    Exercise \(\PageIndex{10}\)

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

    Answer

    75 mph

    Exercise \(\PageIndex{11}\)

    Aisha took the train from Spokane to Seattle. The distance is 280 miles and the trip took 3.5 hours. What was the speed of the train?

    Exercise \(\PageIndex{12}\)

    Philip got a ride with a friend from Denver to Las Vegas, a distance of 750 miles. If the trip took 10 hours, how fast was the friend driving?

    Answer

    75 mph

    Solve a Formula for a Specific Variable

    In the following exercises, use the formula d=rt.

    Exercise \(\PageIndex{12}\)

    Solve for t

    1. when d=350 and r=70
    2. in general

    Exercise \(\PageIndex{13}\)

    Solve for t

    1. when d=240 and r=60
    2. in general
    Answer
    1. t=4
    2. \(t=\frac{d}{r}\)

    Exercise \(\PageIndex{14}\)

    Solve for t

    1. when d=510 and r=60
    2. in general

    Exercise \(\PageIndex{15}\)

    Solve for t

    1. when d=175 and r=50
    2. in general
    Answer
    1. t=3.5
    2. \(t=\frac{d}{r}\)

    Exercise \(\PageIndex{16}\)

    Solve for r

    1. when d=204 and t=3
    2. in general

    Exercise \(\PageIndex{17}\)

    Solve for r

    1. when d=420 and t=6
    2. in general
    Answer
    1. r=70
    2. \(r=\frac{d}{t}\)

    Exercise \(\PageIndex{18}\)

    Solve for r

    1. when d=160 and t=2.5
    2. in general

    Exercise \(\PageIndex{19}\)

    Solve for r

    1. when d=180 and t=4.5
    2. in general
    Answer
    1. r=40
    2. \(r=\frac{d}{t}\)

    In the following exercises, use the formula \(A=\frac{1}{2} b h\)

    Exercise \(\PageIndex{20}\)

    Solve for b

    1. when A=126 and h=18
    2. in general

    Exercise \(\PageIndex{21}\)

    Solve for h

    1. when A=176 and b=22
    2. in general
    Answer
    1. h=16
    2. \(h=\frac{2 A}{b}\)

    Exercise \(\PageIndex{22}\)

    Solve for h

    1. when A=375 and b=25
    2. in general

    Exercise \(\PageIndex{23}\)

    Solve for b

    1. when A=65 and h=13
    2. in general
    Answer
    1. b=10
    2. \(b=\frac{2 A}{h}\)

    In the following exercises, use the formula I = Prt.

    Exercise \(\PageIndex{24}\)

    Solve for the principal, P for

    1. I=$5,480, r=4%, t=7 years
    2. in general

    Exercise \(\PageIndex{25}\)

    Solve for the principal, P for

    1. I=$3,950,r=6%, t=5 years
    2. in general
    Answer
    1. \(P=\$ 13,166.67\)
    2. \( P=\frac{I}{r t}\)

    Exercise \(\PageIndex{26}\)

    Solve for the time, t for

    1. I=$2,376,P=$9,000, r=4.4%
    2. in general

    Exercise \(\PageIndex{27}\)

    Solve for the time, t for

    1. I=$624,P=$6,000, r=5.2%
    2. in general
    Answer
    1. t=2 years
    2. \(t=\frac{I}{\operatorname{Pr}}\)

    In the following exercises, solve.

    Exercise \(\PageIndex{28}\)

    Solve the formula 2x+3y=12 for y

    1. when x=3
    2. in general

    Exercise \(\PageIndex{29}\)

    Solve the formula 5x+2y=10 for y

    1. when x=4
    2. in general
    Answer
    1. y=−5
    2. \(y=\frac{10-5 x}{2}\)

    Exercise \(\PageIndex{30}\)

    Solve the formula 3x−y=7 for y

    1. when x=−2
    2. in general

    Exercise \(\PageIndex{31}\)

    Solve the formula 4x+y=5 for y

    1. when x=−3
    2. in general
    Answer
    1. y=17
    2. y=5−4x

    Exercise \(\PageIndex{32}\)

    Solve a+b=90 for b.

    Exercise \(\PageIndex{33}\)

    Solve \(a+b=90\) for \(a\)

    Answer

    \(a=90-b\)

    Exercise \(\PageIndex{34}\)

    Solve \(180=a+b+c\) for \(a\)

    Exercise \(\PageIndex{35}\)

    Solve \(180=a+b+c\) for \(c\)

    Answer

    \(c=180-a-b\)

    Exercise \(\PageIndex{36}\)

    Solve the formula \(8 x+y=15\) for \(y\)

    Exercise \(\PageIndex{37}\)

    Solve the formula \(9 x+y=13\) for \(y\)

    Answer

    \(y=13-9 x\)

    Exercise \(\PageIndex{38}\)

    Solve the formula \(-4 x+y=-6\) for \(y\)

    Exercise \(\PageIndex{39}\)

    Solve the formula \(-5 x+y=-1\) for \(y\)

    Answer

    \(y=-1+5 x\)

    Exercise \(\PageIndex{40}\)

    Solve the formula \(4 x+3 y=7\) for \(y\)

    Exercise \(\PageIndex{41}\)

    Solve the formula \(3 x+2 y=11\) for \(y\)

    Answer

    \(y=\frac{11-3 x}{4}\)

    Exercise \(\PageIndex{42}\)

    Solve the formula \(x-y=-4\) for \(y\)

    Exercise \(\PageIndex{43}\)

    Solve the formula \(x-y=-3\) for \(y\)

    Answer

    \(y=3+x\)

    Exercise \(\PageIndex{44}\)

    Solve the formula \(P=2 L+2 W\) for \(L\)

    Exercise \(\PageIndex{45}\)

    Solve the formula \(P=2 L+2 W\) for \(W\)

    Answer

    \(W=\frac{P-2 L}{2}\)

    Exercise \(\PageIndex{46}\)

    Solve the formula \(C=\pi d\) for \(d\)

    Exercise \(\PageIndex{47}\)

    Solve the formula \(C=\pi d\) for \(\pi\)

    Answer

    \(\pi=\frac{C}{d}\)

    Exercise \(\PageIndex{48}\)

    Solve the formula \(V=L W H\) for \(L\)

    Exercise \(\PageIndex{49}\)

    Solve the formula \(V=L W H\) for \(H\)

    Answer

    \(H=\frac{V}{L W}\)

    Everyday Math

    Exercise \(\PageIndex{50}\)

    Converting temperature While on a tour in Greece, Tatyana saw that the temperature was 40o Celsius. Solve for F in the formula \(C=\frac{5}{9}(F−32)\) to find the Fahrenheit temperature.

    Exercise \(\PageIndex{51}\)

    Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle one day was 50oFahrenheit. Solve for C in the formula \(F=\frac{9}{5}C+32\) to find the Celsius temperature.

    Answer

    \(10^{\circ} \mathrm{C}\)

    Writing Exercises

    Exercise \(\PageIndex{52}\)

    Solve the equation 2x+3y=6 for y

    1. when x=−3
    2. in general
    3. Which solution is easier for you, 1 or 2? Why?

    Exercise \(\PageIndex{53}\)

    Solve the equation 5x−2y=10 for x

    1. when y=10
    2. in general
    3. Which solution is easier for you, 1 or 2? Why?
    Answer

    Answers will vary.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “use the distance, rate, and time formula,” and “solve a formula for a specific variable.” The rest of the cells are blank.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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