Literal Equations Worksheets - Math Monks - Free Printable

Let's solve each of the literal equations for the specified variable, step by step.

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① $ A = 2\pi rh + 2\pi r^2 $, for $ \pi $

We want to isolate $ \pi $. Notice that $ \pi $ is a common factor in both terms on the right-hand side.

$$
A = 2\pi rh + 2\pi r^2
$$

Factor out $ 2\pi r $:

$$
A = 2\pi r(h + r)
$$

Now divide both sides by $ 2r(h + r) $:

$$
\pi = \frac{A}{2r(h + r)}
$$

✔ Answer:
$$
\boxed{\pi = \frac{A}{2r(h + r)}}
$$

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② $ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} $, for $ T_1 $

We want to solve for $ T_1 $. Start by multiplying both sides by $ T_1 $ to eliminate the denominator:

$$
P_1 V_1 = \frac{P_2 V_2}{T_2} \cdot T_1
$$

Now divide both sides by $ \frac{P_2 V_2}{T_2} $, or multiply both sides by $ \frac{T_2}{P_2 V_2} $:

$$
T_1 = \frac{P_1 V_1 T_2}{P_2 V_2}
$$

✔ Answer:
$$
\boxed{T_1 = \frac{P_1 V_1 T_2}{P_2 V_2}}
$$

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③ $ \frac{1}{f} = \frac{1}{a} + \frac{1}{b} $, for $ f $

We need to solve for $ f $. First, combine the right-hand side:

$$
\frac{1}{f} = \frac{b + a}{ab}
$$

Now take the reciprocal of both sides:

$$
f = \frac{ab}{a + b}
$$

✔ Answer:
$$
\boxed{f = \frac{ab}{a + b}}
$$

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④ $ Q = \frac{11}{5}(P - 15) $, for $ P $

We want to solve for $ P $. Start by multiplying both sides by 5 to eliminate the denominator:

$$
5Q = 11(P - 15)
$$

Now divide both sides by 11:

$$
\frac{5Q}{11} = P - 15
$$

Add 15 to both sides:

$$
P = \frac{5Q}{11} + 15
$$

✔ Answer:
$$
\boxed{P = \frac{5Q}{11} + 15}
$$

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⑤ $ E = mc^2 $, for $ c $

We want to solve for $ c $. Divide both sides by $ m $:

$$
\frac{E}{m} = c^2
$$

Now take the square root of both sides:

$$
c = \sqrt{\frac{E}{m}}
$$

(Note: Since $ c $ represents speed, we usually take the positive root.)

✔ Answer:
$$
\boxed{c = \sqrt{\frac{E}{m}}}
$$

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⑥ $ A = P(1 + rt) $, for $ t $

We want to solve for $ t $. Start by dividing both sides by $ P $:

$$
\frac{A}{P} = 1 + rt
$$

Subtract 1 from both sides:

$$
\frac{A}{P} - 1 = rt
$$

Now divide both sides by $ r $:

$$
t = \frac{\frac{A}{P} - 1}{r} = \frac{A - P}{Pr}
$$

✔ Answer:
$$
\boxed{t = \frac{A - P}{Pr}}
$$

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⑦ $ h = vt - 16t^2 $, for $ v $

We want to solve for $ v $. Isolate the term with $ v $:

$$
h + 16t^2 = vt
$$

Now divide both sides by $ t $ (assuming $ t \neq 0 $):

$$
v = \frac{h + 16t^2}{t}
$$

You can also write this as:

$$
v = \frac{h}{t} + 16t
$$

✔ Answer:
$$
\boxed{v = \frac{h}{t} + 16t}
$$

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⑧ $ m = \frac{y_2 - y_1}{x_2 - x_1} $, for $ y_2 $

We want to solve for $ y_2 $. Multiply both sides by $ x_2 - x_1 $:

$$
m(x_2 - x_1) = y_2 - y_1
$$

Now add $ y_1 $ to both sides:

$$
y_2 = m(x_2 - x_1) + y_1
$$

✔ Answer:
$$
\boxed{y_2 = m(x_2 - x_1) + y_1}
$$

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✔ Final Answers Summary:

1. $ \pi = \dfrac{A}{2r(h + r)} $
2. $ T_1 = \dfrac{P_1 V_1 T_2}{P_2 V_2} $
3. $ f = \dfrac{ab}{a + b} $
4. $ P = \dfrac{5Q}{11} + 15 $
5. $ c = \sqrt{\dfrac{E}{m}} $
6. $ t = \dfrac{A - P}{Pr} $
7. $ v = \dfrac{h}{t} + 16t $
8. $ y_2 = m(x_2 - x_1) + y_1 $

Let me know if you'd like these steps explained further!