Answer :
To solve this problem, we need to figure out how many copies of each type of book were shipped. Let's break it down step by step.
1. Setting up the variables:
- Let [tex]\( x \)[/tex] be the number of copies of the first type of paperback book.
- Let [tex]\( y \)[/tex] be the number of copies of the second type of paperback book.
2. First equation based on the total number of books:
- We know there are a total of 179 books, so we can write:
[tex]\[
x + y = 179
\][/tex]
3. Second equation based on the total weight of the books:
- Each copy of the first type of paperback weighs [tex]\(\frac{2}{3}\)[/tex] of a pound.
- Each copy of the second type of paperback weighs [tex]\(\frac{3}{4}\)[/tex] of a pound.
- The total weight of all the books is 128 pounds. This gives us the equation:
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
4. Checking the system of equations:
- The correct system of equations is:
[tex]\[
x + y = 179
\][/tex]
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
5. Solving the equations:
- You can use substitution or elimination methods to solve this system. For elimination, it might be easier to first clear the fractions by finding a common multiple or by simply solving as is using algebraic methods.
6. Solution:
- Solving these equations, we find that there are:
- 75 copies of the first paperback book.
- 104 copies of the second paperback book.
With this solution, we can confirm which statements about the problem are true:
- The statement "The system of equations is [tex]\(x + y = 179\)[/tex] and [tex]\(\frac{2}{3}x + \frac{3}{4}y = 128\)[/tex]" is true.
- The statement "The system of equations is [tex]\(x + y = 128\)[/tex] and [tex]\(\frac{2}{3}x + \frac{3}{4}y = 179\)[/tex]" is false.
- The statement about elimination methods is not directly verifiable without conducting elimination steps.
- The statement "There are 104 copies of one book and 24 copies of the other" is false, per the solution, which indicated there are 75 copies of one book and 104 copies of the other.
These are the conclusions drawn from solving the problem.
1. Setting up the variables:
- Let [tex]\( x \)[/tex] be the number of copies of the first type of paperback book.
- Let [tex]\( y \)[/tex] be the number of copies of the second type of paperback book.
2. First equation based on the total number of books:
- We know there are a total of 179 books, so we can write:
[tex]\[
x + y = 179
\][/tex]
3. Second equation based on the total weight of the books:
- Each copy of the first type of paperback weighs [tex]\(\frac{2}{3}\)[/tex] of a pound.
- Each copy of the second type of paperback weighs [tex]\(\frac{3}{4}\)[/tex] of a pound.
- The total weight of all the books is 128 pounds. This gives us the equation:
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
4. Checking the system of equations:
- The correct system of equations is:
[tex]\[
x + y = 179
\][/tex]
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
5. Solving the equations:
- You can use substitution or elimination methods to solve this system. For elimination, it might be easier to first clear the fractions by finding a common multiple or by simply solving as is using algebraic methods.
6. Solution:
- Solving these equations, we find that there are:
- 75 copies of the first paperback book.
- 104 copies of the second paperback book.
With this solution, we can confirm which statements about the problem are true:
- The statement "The system of equations is [tex]\(x + y = 179\)[/tex] and [tex]\(\frac{2}{3}x + \frac{3}{4}y = 128\)[/tex]" is true.
- The statement "The system of equations is [tex]\(x + y = 128\)[/tex] and [tex]\(\frac{2}{3}x + \frac{3}{4}y = 179\)[/tex]" is false.
- The statement about elimination methods is not directly verifiable without conducting elimination steps.
- The statement "There are 104 copies of one book and 24 copies of the other" is false, per the solution, which indicated there are 75 copies of one book and 104 copies of the other.
These are the conclusions drawn from solving the problem.
