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Answer :
To solve this problem, we need to find out how many copies of each type of paperback book are there. Let's go through the details step-by-step based on the given system of equations:
1. Identify 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. Set Up Equations:
- The first equation from the total number of books is:
[tex]\[
x + y = 179
\][/tex]
- The second equation comes from the total weight of the books:
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
3. Solve the System of Equations:
- Equation 1: [tex]\( x + y = 179 \)[/tex]
- Equation 2: [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex]
4. Eliminate One Variable:
- To eliminate a variable, manipulate one or both of the equations, but note that in problems involving fractions like these, simplifying or clearing the fractions can help streamline calculations.
5. Solve for One Variable and Substitute:
- From Equation 1, express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = 179 - y
\][/tex]
- Substitute [tex]\( x = 179 - y \)[/tex] into Equation 2:
[tex]\[
\frac{2}{3}(179 - y) + \frac{3}{4}y = 128
\][/tex]
6. Clear Fractions and Solve:
- Multiply through by a common multiple of the denominators to eliminate the fractions. For example, multiply the entire equation by 12 to clear [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex].
7. Continue Solving:
- Simplify and solve the resulting equation to find [tex]\( y \)[/tex].
8. Find [tex]\( x \)[/tex]:
- Substitute the value of [tex]\( y \)[/tex] back into the expression [tex]\( x = 179 - y \)[/tex] to find [tex]\( x \)[/tex].
Given the calculation results from the solution, you find:
- [tex]\( x = 75 \)[/tex]: There are 75 copies of the first type of paperback book.
- [tex]\( y = 104 \)[/tex]: There are 104 copies of the second type of paperback book.
Conclusion:
- The statement about there being 104 copies of one book is true.
- The system set up for the problem initially was correct with [tex]\( x + y = 179 \)[/tex] and [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex].
- The option regarding multiplying to eliminate variables also follows standard algebraic processes for solving systems involving fractions.
1. Identify 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. Set Up Equations:
- The first equation from the total number of books is:
[tex]\[
x + y = 179
\][/tex]
- The second equation comes from the total weight of the books:
[tex]\[
\frac{2}{3}x + \frac{3}{4}y = 128
\][/tex]
3. Solve the System of Equations:
- Equation 1: [tex]\( x + y = 179 \)[/tex]
- Equation 2: [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex]
4. Eliminate One Variable:
- To eliminate a variable, manipulate one or both of the equations, but note that in problems involving fractions like these, simplifying or clearing the fractions can help streamline calculations.
5. Solve for One Variable and Substitute:
- From Equation 1, express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = 179 - y
\][/tex]
- Substitute [tex]\( x = 179 - y \)[/tex] into Equation 2:
[tex]\[
\frac{2}{3}(179 - y) + \frac{3}{4}y = 128
\][/tex]
6. Clear Fractions and Solve:
- Multiply through by a common multiple of the denominators to eliminate the fractions. For example, multiply the entire equation by 12 to clear [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex].
7. Continue Solving:
- Simplify and solve the resulting equation to find [tex]\( y \)[/tex].
8. Find [tex]\( x \)[/tex]:
- Substitute the value of [tex]\( y \)[/tex] back into the expression [tex]\( x = 179 - y \)[/tex] to find [tex]\( x \)[/tex].
Given the calculation results from the solution, you find:
- [tex]\( x = 75 \)[/tex]: There are 75 copies of the first type of paperback book.
- [tex]\( y = 104 \)[/tex]: There are 104 copies of the second type of paperback book.
Conclusion:
- The statement about there being 104 copies of one book is true.
- The system set up for the problem initially was correct with [tex]\( x + y = 179 \)[/tex] and [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex].
- The option regarding multiplying to eliminate variables also follows standard algebraic processes for solving systems involving fractions.
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