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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.
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.
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