High School

A box containing a total of 179 copies of two different paperback books was shipped to Marci's school. The total weight of the books was 128 pounds. If the weight of each of the first paperbacks was [tex]\frac{2}{3}[/tex] of a pound and the weight of each of the second paperbacks was [tex]\frac{3}{4}[/tex] of a pound, which statements are true? Check all that apply.

- The system of equations is [tex]x+y=179[/tex] and [tex]\frac{2}{3}x+\frac{3}{4}y=128[/tex].
- The system of equations is [tex]x+y=128[/tex] and [tex]\frac{2}{3}x+\frac{3}{4}y=179[/tex].
- To eliminate the [tex]x[/tex]-variable from the equations, you can multiply the equation with the fractions by 3 and leave the other equation as it is.
- To eliminate the [tex]y[/tex]-variable from the equations, you can multiply the equation with the fractions by -4 and multiply the other equation by 3.
- There are 104 copies of one book and 24 copies of the other.

Answer :

To solve the problem, we need to figure out how many copies of each book Marci received. Let's go step by step:

1. Set Up the Equations:
- We know there are a total of 179 copies of books. So, we can set up the first equation:
[tex]\[
x + y = 179
\][/tex]
Here, [tex]\(x\)[/tex] is the number of the first type of paperback book, and [tex]\(y\)[/tex] is the number of the second type.

- The total weight of books is 128 pounds. Each first paperback weighs [tex]\(\frac{2}{3}\)[/tex] of a pound, and each second paperback weighs [tex]\(\frac{3}{4}\)[/tex] of a pound. Therefore, the second equation is:
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]

2. Analyze the System of Equations:
- The provided system of equations to work with is:
[tex]\[
x + y = 179
\][/tex]
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]

3. Solving the Equations:
- The system of equations suggests that by substituting or eliminating variables, we can find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- Let's multiply the entire second equation to eliminate the fractions. Multiply by 12 (LCM of 3 and 4):
[tex]\[
12 \left(\frac{2}{3}x + \frac{3}{4}y\right) = 12 \times 128
\][/tex]
Simplifying,
[tex]\[
8x + 9y = 1536
\][/tex]

- Now, solve the system:
1. Multiply the first equation by 8 (to eliminate [tex]\(x\)[/tex] using elimination method) and keep the second equation as it is:
[tex]\[
8(x + y) = 8 \times 179
\][/tex]
Simplifying:
[tex]\[
8x + 8y = 1432
\][/tex]
2. Now, subtract the modified first equation from the modified second:
[tex]\[
(8x + 9y) - (8x + 8y) = 1536 - 1432
\][/tex]
[tex]\[
y = 104
\][/tex]
- Substitute [tex]\(y = 104\)[/tex] back into the first equation:
[tex]\[
x + 104 = 179
\][/tex]
[tex]\[
x = 75
\][/tex]

4. Conclusion:
- [tex]\(x = 75\)[/tex] and [tex]\(y = 104\)[/tex], meaning there are 75 copies of the first type of paperback book and 104 copies of the second type.

5. Verify the Statements:
- The correct system of equations is indeed [tex]\(x + y = 179\)[/tex] and [tex]\(\frac{2}{3}x + \frac{3}{4}y = 128\)[/tex].
- The values of [tex]\(x = 75\)[/tex] and [tex]\(y = 104\)[/tex] satisfy the condition for the total number of books and the total weight of the books.
- Hence, statements regarding the system of equations and the solutions are correct, and there are 75 copies of one book and 104 copies of the other.