Algebra 1 - Mr. C. - Practice B - BulldogMath.com - Free Printable

Let's solve each problem step by step from the worksheet titled "Algebra 1 - Mr. C. - Practice B: Solving for a Variable".

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1. The formula $ C = 2\pi r $ relates the radius $ r $ of a circle to its circumference $ C $. Solve this formula for $ r $.

We start with:
$$
C = 2\pi r
$$

To solve for $ r $, divide both sides by $ 2\pi $:
$$
r = \frac{C}{2\pi}
$$

✔ Answer: $ r = \frac{C}{2\pi} $

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2. The formula $ y = mx + b $ is called the slope-intercept form of a line. Solve this formula for $ m $.

Start with:
$$
y = mx + b
$$

Subtract $ b $ from both sides:
$$
y - b = mx
$$

Now divide both sides by $ x $ (assuming $ x \neq 0 $):
$$
m = \frac{y - b}{x}
$$

✔ Answer: $ m = \frac{y - b}{x} $

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Solve for the indicated variable:

#### 3. $ 4c = d $ for $ c $

Divide both sides by 4:
$$
c = \frac{d}{4}
$$

✔ Answer: $ c = \frac{d}{4} $

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#### 4. $ n - 6m = 8 $ for $ n $

Add $ 6m $ to both sides:
$$
n = 8 + 6m
$$

✔ Answer: $ n = 6m + 8 $

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#### 5. $ 2p + 5r = q $ for $ p $

Subtract $ 5r $ from both sides:
$$
2p = q - 5r
$$

Divide by 2:
$$
p = \frac{q - 5r}{2}
$$

✔ Answer: $ p = \frac{q - 5r}{2} $

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#### 6. $ -10 = xy + z $ for $ x $

Subtract $ z $ from both sides:
$$
-10 - z = xy
$$

Now divide both sides by $ y $ (assuming $ y \neq 0 $):
$$
x = \frac{-10 - z}{y}
$$

✔ Answer: $ x = \frac{-10 - z}{y} $

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#### 7. $ \frac{a}{b} = c $ for $ b $

Multiply both sides by $ b $:
$$
a = bc
$$

Now divide both sides by $ c $ (assuming $ c \neq 0 $):
$$
b = \frac{a}{c}
$$

✔ Answer: $ b = \frac{a}{c} $

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#### 8. $ \frac{h - 4}{j} = k $ for $ j $

Multiply both sides by $ j $:
$$
h - 4 = kj
$$

Now divide both sides by $ k $ (assuming $ k \neq 0 $):
$$
j = \frac{h - 4}{k}
$$

✔ Answer: $ j = \frac{h - 4}{k} $

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9. The formula $ c = 5p + 215 $ relates $ c $, the total cost in dollars of hosting a birthday party at a skating rink, to $ p $, the number of people attending.

#### a. Solve the formula $ c = 5p + 215 $ for $ p $.

Subtract 215 from both sides:
$$
c - 215 = 5p
$$

Divide by 5:
$$
p = \frac{c - 215}{5}
$$

✔ Answer: $ p = \frac{c - 215}{5} $

#### b. If Allie’s parents are willing to spend $300 for a party, how many people can attend?

Use the formula from part (a) with $ c = 300 $:
$$
p = \frac{300 - 215}{5} = \frac{85}{5} = 17
$$

✔ Answer: 17 people can attend.

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10. The formula for the area of a triangle is $ A = \frac{1}{2}bh $, where $ b $ is the base and $ h $ is the height.

#### a. Solve the formula $ A = \frac{1}{2}bh $ for $ b $.

Multiply both sides by 2:
$$
2A = bh
$$

Now divide both sides by $ h $ (assuming $ h \neq 0 $):
$$
b = \frac{2A}{h}
$$

✔ Answer: $ b = \frac{2A}{h} $

#### b. If a triangle has an area of $ 192 \, \text{mm}^2 $, and the height measures $ 12 \, \text{mm} $, what is the measure of the base?

Use the formula from part (a):
$$
b = \frac{2 \times 192}{12} = \frac{384}{12} = 32
$$

✔ Answer: The base is $ 32 \, \text{mm} $.

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✔ Final Answers Summary:

1. $ r = \frac{C}{2\pi} $
2. $ m = \frac{y - b}{x} $
3. $ c = \frac{d}{4} $
4. $ n = 6m + 8 $
5. $ p = \frac{q - 5r}{2} $
6. $ x = \frac{-10 - z}{y} $
7. $ b = \frac{a}{c} $
8. $ j = \frac{h - 4}{k} $
9a. $ p = \frac{c - 215}{5} $
9b. 17 people
10a. $ b = \frac{2A}{h} $
10b. 32 mm

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