Answer :
Let's break down the problem and work through the information given step-by-step to understand which statements are true:
1. Understanding the Problem:
- We have a total of 179 books, which consist of two different types of paperback books.
- The total weight of all these books is 128 pounds.
- Each of the first type of paperback weighs [tex]\(\frac{2}{3}\)[/tex] of a pound.
- Each of the second type of paperback weighs [tex]\(\frac{3}{4}\)[/tex] of a pound.
2. Setting Up the Equations:
- Let [tex]\( x \)[/tex] represent the number of the first type of book.
- Let [tex]\( y \)[/tex] represent the number of the second type of book.
From the information given, we can set up the following equations:
- Equation 1 (total books): [tex]\( x + y = 179 \)[/tex]
- Equation 2 (total weight): [tex]\(\frac{2}{3}x + \frac{3}{4}y = 128\)[/tex]
3. Solving the System of Equations:
- We have a system of linear equations, and we need to solve it to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
After solving these equations, we find:
- [tex]\( x = 75 \)[/tex]
- [tex]\( y = 104 \)[/tex]
4. Analyzing the Statements:
- Let's evaluate the statements based on the solutions:
- The system of equations is [tex]\(x+y=179\)[/tex] and [tex]\(\frac{2}{3} x+\frac{3}{4} y=128\)[/tex]. True. These are indeed the correct equations based on the problem.
- The system of equations is [tex]\(x+y=128\)[/tex] and [tex]\(\frac{2}{3} x+\frac{3}{4} y=179\)[/tex]. False. This statement has the equations reversed.
- 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. True. Multiplying the fractions equation for simplification is a valid step in elimination.
- To eliminate the [tex]\(y\)[/tex]-variable from the equations, you can multiply the equation with the fractions by [tex]\(-4\)[/tex] and multiply the other equation by 3. This statement is unclear because multiplying by [tex]\(-4\)[/tex] specifically needs more context about this operation.
- There are 104 copies of one book and 24 copies of the other. False. According to the solution, there are 75 copies of one book and 104 copies of the other.
Hence, with the information and solutions provided, only the statements about the correct formulation of equations and potential steps for elimination are true, along with the specific values found for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Understanding the Problem:
- We have a total of 179 books, which consist of two different types of paperback books.
- The total weight of all these books is 128 pounds.
- Each of the first type of paperback weighs [tex]\(\frac{2}{3}\)[/tex] of a pound.
- Each of the second type of paperback weighs [tex]\(\frac{3}{4}\)[/tex] of a pound.
2. Setting Up the Equations:
- Let [tex]\( x \)[/tex] represent the number of the first type of book.
- Let [tex]\( y \)[/tex] represent the number of the second type of book.
From the information given, we can set up the following equations:
- Equation 1 (total books): [tex]\( x + y = 179 \)[/tex]
- Equation 2 (total weight): [tex]\(\frac{2}{3}x + \frac{3}{4}y = 128\)[/tex]
3. Solving the System of Equations:
- We have a system of linear equations, and we need to solve it to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
After solving these equations, we find:
- [tex]\( x = 75 \)[/tex]
- [tex]\( y = 104 \)[/tex]
4. Analyzing the Statements:
- Let's evaluate the statements based on the solutions:
- The system of equations is [tex]\(x+y=179\)[/tex] and [tex]\(\frac{2}{3} x+\frac{3}{4} y=128\)[/tex]. True. These are indeed the correct equations based on the problem.
- The system of equations is [tex]\(x+y=128\)[/tex] and [tex]\(\frac{2}{3} x+\frac{3}{4} y=179\)[/tex]. False. This statement has the equations reversed.
- 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. True. Multiplying the fractions equation for simplification is a valid step in elimination.
- To eliminate the [tex]\(y\)[/tex]-variable from the equations, you can multiply the equation with the fractions by [tex]\(-4\)[/tex] and multiply the other equation by 3. This statement is unclear because multiplying by [tex]\(-4\)[/tex] specifically needs more context about this operation.
- There are 104 copies of one book and 24 copies of the other. False. According to the solution, there are 75 copies of one book and 104 copies of the other.
Hence, with the information and solutions provided, only the statements about the correct formulation of equations and potential steps for elimination are true, along with the specific values found for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
