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Define a variable and write an equation for each situation. Then solve.

21. **Donations**
A library receives a large cash donation and uses the funds to double the number of books it owns. Then a book collector gives the library 4028 books. After this, the library has 51,514 books. How many books did the library have before the cash donation and the gift of books?

22. **Cooking**
Suppose you are helping to prepare a large meal. You can peel 2 carrots per minute. You need 60 peeled carrots. How long will it take you to finish if you have already peeled 18 carrots?

23. **Cell Phones**
One cell phone plan costs [tex]$\$39.95$[/tex] per month. The first 500 minutes of usage are free. Each minute thereafter costs [tex]$\$0.35$[/tex]. Write a rule that describes the total monthly cost as a function of the number of minutes of usage (over 500 minutes). Then find the number of minutes of usage over 500 minutes for a bill of [tex]$\$69.70$[/tex].

24. Justify each step.
\[
\begin{aligned}
\frac{x}{5}+9 & =11 \\
\frac{x}{5}+9-9 & =11-9 \\
\frac{x}{5} & =2 \\
5\left(\frac{x}{5}\right) & =5(2) \\
x & =10
\end{aligned}
\]

25.
\[
\begin{aligned}
-y-5 & =11 \\
-y-5+5 & =11+5 \\
-y & =16 \\
-1(-y) & =-1(16) \\
y & =-16
\end{aligned}
\]

26.
\[
\begin{aligned}
18-n & =21 \\
18-n-18 & =21-18 \\
-n & =3 \\
-1(-n) & =-1(3) \\
n & =-3
\end{aligned}
\]

27.
\[
\begin{aligned}
12-2h & =8 \\
12-2h-12 & =8-12 \\
-2h & =-4 \\
\frac{-2h}{-2} & =\frac{-4}{-2} \\
h & =2
\end{aligned}
\]

Answer :

Sure, let's solve each problem step-by-step.

### 21. Donations:
Situation: The library receives a cash donation which they use to double the number of books they initially had. Then, a book collector donates 4028 books. After these additions, the library has a total of 51,514 books.
Goal: Find the initial number of books the library had before any donations.

1. Let [tex]\( x \)[/tex] be the initial number of books the library had.
2. After doubling the number of books, the library has [tex]\( 2x \)[/tex] books.
3. After receiving an additional 4028 books, they have [tex]\( 2x + 4028 \)[/tex] books.
4. We know this sum equals 51,514 books.

So, the equation is:
[tex]\[ 2x + 4028 = 51514 \][/tex]

Subtract 4028 from both sides:
[tex]\[ 2x = 47486 \][/tex]

Divide by 2:
[tex]\[ x = 23743 \][/tex]

Therefore, the library initially had 23,743 books.

### 22. Cooking:
Situation: You can peel 2 carrots per minute. You need 60 peeled carrots and you've already peeled 18 carrots.
Goal: Find how long it will take to peel the remaining carrots.

1. Calculate the number of carrots left to peel:
[tex]\[ \text{Needed carrots} = 60 - 18 = 42 \][/tex]

2. Since you can peel 2 carrots per minute, the time needed to peel 1 carrot is:
[tex]\[ \text{Time per carrot} = \frac{1}{2} \text{ minutes} \][/tex]

3. Multiply the remaining carrots by the time per carrot to find the total time needed:
[tex]\[ \text{Total time} = 42 \times \frac{1}{2} = 21 \text{ minutes} \][/tex]

So, it will take you 21 minutes to finish peeling the remaining carrots.

### 23. Cell Phones:
Situation: One cell phone plan costs \[tex]$39.95 per month. The first 500 minutes are free, and each minute over 500 costs \$[/tex]0.35.
Goal: Write a rule for the total monthly cost as a function of the minutes over 500, and find the number of over-500 minutes if the total bill is \[tex]$69.70.

1. Let \( x \) be the number of minutes used over 500.
2. The cost for the minutes over 500 is \( 0.35x \).
3. The total monthly cost is then \( 39.95 + 0.35x \).

The given total monthly bill is \$[/tex]69.70:
[tex]\[ 39.95 + 0.35x = 69.70 \][/tex]

Subtract 39.95 from both sides:
[tex]\[ 0.35x = 29.75 \][/tex]

Divide by 0.35:
[tex]\[ x = 85 \][/tex]

Therefore, the number of minutes used over 500 is 85 minutes.

### 24. Solving Equations:
Given steps:
[tex]\[ \begin{aligned}
\frac{x}{5} + 9 &= 11 \\
\frac{x}{5} + 9 - 9 &= 11 - 9 \\
\frac{x}{5} &= 2 \\
5 \cdot \frac{x}{5} &= 5 \cdot 2 \\
x &= 10
\end{aligned} \][/tex]

So the value of [tex]\( x \)[/tex] is 10.

### 25. Solving Equations:
Given steps:
[tex]\[ \begin{aligned}
-y - 5 &= 11 \\
-y - 5 + 5 &= 11 + 5 \\
-y &= 16 \\
-1(-y) &= -1(16) \\
y &= -16
\end{aligned} \][/tex]

So the value of [tex]\( y \)[/tex] is -16.

### 26. Solving Equations:
Given steps:
[tex]\[ \begin{aligned}
18 - n &= 21 \\
18 - n - 18 &= 21 - 18 \\
-n &= 3 \\
-1(-n) &= -1(3) \\
n &= -3
\end{aligned} \][/tex]

So the value of [tex]\( n \)[/tex] is -3.

### 27. Solving Equations:
Given steps:
[tex]\[ \begin{aligned}
12 - 2h &= 8 \\
12 - 2h - 12 &= 8 - 12 \\
-2h &= -4 \\
\frac{-2h}{-2} &= \frac{-4}{-2} \\
h &= 2
\end{aligned} \][/tex]

So the value of [tex]\( h \)[/tex] is 2.

These are the step-by-step solutions to the given problems. If you need further assistance, feel free to ask!

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