Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 12

Worked example (no skipped steps)

12.1 Use 𝐴 = 𝑙𝑤 with 𝑙 = 8 and 𝑤 = 3.

Substitute the values:

𝐴 = (8)(3)

Multiply:

𝐴 = 24

Answer: 24.

Worked example (no skipped steps)

12.2 Use 𝑑 = 𝑟𝑡 with 𝑟 = 60 and 𝑡 = 2 . 5.

Substitute:

𝑑 = (60)(2 . 5)

Multiply:

𝑑 = 150

Answer: 150.

Worked example (no skipped steps)

12.3 Use 𝑦 = 𝑚𝑥 + 𝑏 with 𝑚 = 2, 𝑥 = 5, 𝑏 = −3.

Substitute carefully:

𝑦 = (2)(5) + (−3)

Now simplify:

𝑦 = 10 − 3 = 7

Answer: 7.

184

Step-by-step solutions

Worked example (no skipped steps)

12.4 Solve for 𝑏 in 𝑦 = 𝑚𝑥 + 𝑏.

Subtract 𝑚𝑥 from both sides:

𝑦 − 𝑚𝑥 = 𝑏

So:

𝑏 = 𝑦 − 𝑚𝑥

Answer: 𝑏 = 𝑦 − 𝑚𝑥.

Worked example (no skipped steps)

12.5 Solve for 𝑡 in 𝑑 = 𝑟𝑡.

Divide both sides by 𝑟:

𝑑

𝑟𝑡

=

𝑟

𝑟

So:

𝑑

𝑡 = 𝑟

Answer: 𝑡 = 𝑑 (with 𝑟 ̸= 0).

𝑟

Worked example (no skipped steps)

12.6 Solve for 𝑤 in 𝑃 = 2 𝑙 + 2 𝑤.

Subtract 2 𝑙 from both sides:

𝑃 − 2 𝑙 = 2 𝑤

Now divide by 2:

𝑃 − 2 𝑙

𝑤 =

2

Answer: 𝑤 = 𝑃 −2 𝑙 .

2

Worked example (no skipped steps)

12.7 Solve for 𝐶 in 𝐹 = 9 𝐶 + 32.

5

Subtract 32:

9

𝐹 − 32 =

𝐶

5

Multiply both sides by the reciprocal 5 :

9

5 ( 𝐹 − 32) = 𝐶

9

So:

5

𝐶 =

( 𝐹 − 32)

9

Answer: 𝐶 = 5 ( 𝐹 − 32).

9

Chapter 12

185

Worked example (no skipped steps)

12.8 Solve for 𝑟 in 𝐴 = 𝜋𝑟 2.

Divide by 𝜋:

𝐴 = 𝑟 2

𝜋

Take the square root of both sides:

√︃

𝐴

𝑟 =

𝜋

√︁

Answer: 𝑟 =

𝐴 .

𝜋

Worked example (no skipped steps)

12.9 Solve for 𝑥 in 𝑥− 𝑎 = 𝑐.

𝑏

Multiply both sides by 𝑏:

𝑥 − 𝑎 = 𝑏𝑐

Add 𝑎 to both sides:

𝑥 = 𝑏𝑐 + 𝑎

Answer: 𝑥 = 𝑏𝑐 + 𝑎.

Worked example (no skipped steps)

12.10 Solve for 𝑏 in 𝐴 = 1 𝑏ℎ.

2

Multiply both sides by 2:

2 𝐴 = 𝑏ℎ

Now divide by ℎ:

2 𝐴

𝑏 = ℎ

Answer: 𝑏 = 2 𝐴 (with ℎ ̸= 0).

ℎ

Worked example (no skipped steps)

12.11 Solve for 𝑥 in 𝑧 = 𝑥− 𝑦 .

3

Multiply both sides by 3:

3 𝑧 = 𝑥 − 𝑦

Add 𝑦 to both sides:

𝑥 = 3 𝑧 + 𝑦

Answer: 𝑥 = 3 𝑧 + 𝑦.

186

Step-by-step solutions

Worked example (no skipped steps)

12.12 Use 𝑃 = 2 𝑙 + 2 𝑤 with 𝑙 = 7 and 𝑤 = 4.

Substitute:

𝑃 = 2(7) + 2(4)

Simplify:

𝑃 = 14 + 8 = 22

Answer: 22.

Worked example (no skipped steps)

12.13 Use 𝐼 = 𝑃 𝑟𝑡 with 𝑃 = 500, 𝑟 = 0 . 08, 𝑡 = 2.

Substitute:

𝐼 = (500)(0 . 08)(2)

Multiply step by step:

500 · 0 . 08 = 40

40 · 2 = 80

Answer: 80.

Worked example (no skipped steps)

12.14 Solve for 𝑃 in 𝐼 = 𝑃 𝑟𝑡.

Divide both sides by 𝑟𝑡:

𝐼

𝑃 = 𝑟𝑡

Answer: 𝑃 = 𝐼 (with 𝑟 ̸= 0 and 𝑡 ̸= 0).

𝑟𝑡

Worked example (no skipped steps)

12.15 Solve for ℎ in 𝐴 = 1 𝑏ℎ.

2

Multiply both sides by 2:

2 𝐴 = 𝑏ℎ

Divide both sides by 𝑏:

2 𝐴

ℎ = 𝑏

Answer: ℎ = 2 𝐴 (with 𝑏 ̸= 0).

𝑏

Chapter 12

187

Worked example (no skipped steps)

12.16 Solve for 𝑥 in 𝑦 = 𝑎𝑥 + 𝑏.

Subtract 𝑏:

𝑦 − 𝑏 = 𝑎𝑥

Divide by 𝑎:

𝑦 − 𝑏

𝑥 =

𝑎

Answer: 𝑥 = 𝑦− 𝑏 (with 𝑎 ̸= 0).

𝑎

Worked example (no skipped steps)

12.17 Solve for 𝑟 in 𝐶 = 2 𝜋𝑟.

Divide both sides by 2 𝜋:

𝐶

𝑟 = 2 𝜋

Answer: 𝑟 = 𝐶 .

2 𝜋

Worked example (no skipped steps)

12.18 Use 𝐶 = 2 𝜋𝑟 with 𝑟 = 3.

Substitute:

𝐶 = 2 𝜋(3)

Simplify:

𝐶 = 6 𝜋

Answer: 6 𝜋.

Worked example (no skipped steps)

12.19 Use 𝑠 = 𝑢𝑡 + 1 𝑎𝑡 2 with 𝑢 = 4, 𝑎 = 2, 𝑡 = 3.

2

Substitute:

1

𝑠 = (4)(3) +

(2)(32)

2

Simplify each part:

(4)(3) = 12

32 = 9

1 (2)(9) = 1 · 9 = 9

2

Now add:

𝑠 = 12 + 9 = 21

Answer: 21.

188

Step-by-step solutions

Worked example (no skipped steps)

12.20 Solve for 𝑎 in 𝑠 = 𝑢𝑡 + 1 𝑎𝑡 2.

2

Subtract 𝑢𝑡 from both sides:

1

𝑠 − 𝑢𝑡 =

𝑎𝑡 2

2

Multiply both sides by 2:

2( 𝑠 − 𝑢𝑡) = 𝑎𝑡 2

Now divide by 𝑡 2:

2( 𝑠 − 𝑢𝑡)

𝑎 =

𝑡 2

Answer: 𝑎 = 2( 𝑠− 𝑢𝑡) (with 𝑡 ̸= 0).

𝑡 2