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Answer :
Let's solve this problem step by step:
We are given that there are 179 copies of two different paperback books in total, and the total weight of these books is 128 pounds. The first type of paperback book weighs [tex]\(\frac{2}{3}\)[/tex] of a pound each, and the second type weighs [tex]\(\frac{3}{4}\)[/tex] of a pound each.
To find out how many copies of each book there are, we can set up a system of equations based on the information provided:
1. Let [tex]\(x\)[/tex] be the number of the first type of book.
2. Let [tex]\(y\)[/tex] be the number of the second type of book.
From the problem, we have these two equations:
1. [tex]\(x + y = 179\)[/tex] (Equation 1: Total number of books)
2. [tex]\(\frac{2}{3}x + \frac{3}{4}y = 128\)[/tex] (Equation 2: Total weight of books)
Let's solve these equations:
Step 1: Solve Equation 1 for one variable:
[tex]\[ y = 179 - x \][/tex]
Step 2: Substitute [tex]\(y\)[/tex] in Equation 2:
[tex]\[
\frac{2}{3}x + \frac{3}{4}(179 - x) = 128
\][/tex]
Step 3: Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{2}{3}x + \frac{3}{4} \times 179 - \frac{3}{4}x = 128
\][/tex]
Calculate [tex]\(\frac{3}{4} \times 179\)[/tex]:
[tex]\[
\frac{3}{4} \times 179 = \frac{537}{4} = 134.25
\][/tex]
Now substitute back:
[tex]\[
\frac{2}{3}x + 134.25 - \frac{3}{4}x = 128
\][/tex]
Combine like terms:
Convert the fractions to a common denominator:
The common denominator for 3 and 4 is 12.
[tex]\[
\frac{8}{12}x - \frac{9}{12}x = 128 - 134.25
\][/tex]
Combine like terms:
[tex]\[
-\frac{1}{12}x = -6.25
\][/tex]
Multiply both sides by [tex]\(-12\)[/tex]:
[tex]\[
x = 75
\][/tex]
Step 4: Use [tex]\(x = 75\)[/tex] to find [tex]\(y\)[/tex]:
Using Equation 1:
[tex]\[
y = 179 - x = 179 - 75 = 104
\][/tex]
Now, let's check which statements are true:
1. 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 the equations we used.
2. The system of equations is [tex]\(x+y=128\)[/tex] and [tex]\(\frac{2}{3}x+\frac{3}{4}y=179\)[/tex].
- False: These equations do not match the problem statement.
3. 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 all terms in the fraction equation by 3 helps to clear the denominators related to [tex]\(x\)[/tex].
4. 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.
- True: This approach uses multiplication to clear denominators and align terms for elimination.
5. There are 104 copies of one book and 24 copies of the other.
- False: There are 75 copies of one book and 104 copies of the other.
Hence, the true statements are: 1, 3, and 4.
We are given that there are 179 copies of two different paperback books in total, and the total weight of these books is 128 pounds. The first type of paperback book weighs [tex]\(\frac{2}{3}\)[/tex] of a pound each, and the second type weighs [tex]\(\frac{3}{4}\)[/tex] of a pound each.
To find out how many copies of each book there are, we can set up a system of equations based on the information provided:
1. Let [tex]\(x\)[/tex] be the number of the first type of book.
2. Let [tex]\(y\)[/tex] be the number of the second type of book.
From the problem, we have these two equations:
1. [tex]\(x + y = 179\)[/tex] (Equation 1: Total number of books)
2. [tex]\(\frac{2}{3}x + \frac{3}{4}y = 128\)[/tex] (Equation 2: Total weight of books)
Let's solve these equations:
Step 1: Solve Equation 1 for one variable:
[tex]\[ y = 179 - x \][/tex]
Step 2: Substitute [tex]\(y\)[/tex] in Equation 2:
[tex]\[
\frac{2}{3}x + \frac{3}{4}(179 - x) = 128
\][/tex]
Step 3: Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{2}{3}x + \frac{3}{4} \times 179 - \frac{3}{4}x = 128
\][/tex]
Calculate [tex]\(\frac{3}{4} \times 179\)[/tex]:
[tex]\[
\frac{3}{4} \times 179 = \frac{537}{4} = 134.25
\][/tex]
Now substitute back:
[tex]\[
\frac{2}{3}x + 134.25 - \frac{3}{4}x = 128
\][/tex]
Combine like terms:
Convert the fractions to a common denominator:
The common denominator for 3 and 4 is 12.
[tex]\[
\frac{8}{12}x - \frac{9}{12}x = 128 - 134.25
\][/tex]
Combine like terms:
[tex]\[
-\frac{1}{12}x = -6.25
\][/tex]
Multiply both sides by [tex]\(-12\)[/tex]:
[tex]\[
x = 75
\][/tex]
Step 4: Use [tex]\(x = 75\)[/tex] to find [tex]\(y\)[/tex]:
Using Equation 1:
[tex]\[
y = 179 - x = 179 - 75 = 104
\][/tex]
Now, let's check which statements are true:
1. 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 the equations we used.
2. The system of equations is [tex]\(x+y=128\)[/tex] and [tex]\(\frac{2}{3}x+\frac{3}{4}y=179\)[/tex].
- False: These equations do not match the problem statement.
3. 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 all terms in the fraction equation by 3 helps to clear the denominators related to [tex]\(x\)[/tex].
4. 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.
- True: This approach uses multiplication to clear denominators and align terms for elimination.
5. There are 104 copies of one book and 24 copies of the other.
- False: There are 75 copies of one book and 104 copies of the other.
Hence, the true statements are: 1, 3, and 4.
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