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Answer :
Let's solve this problem step-by-step:
Given:
- 179 copies of books in total.
- Total weight is 128 pounds.
- Each of the first books weighs [tex]\( \frac{2}{3} \)[/tex] pounds.
- Each of the second books weighs [tex]\( \frac{3}{4} \)[/tex] pounds.
We need to verify the given statements.
1. Formulate the system of equations.
The problem gives us two types of equations:
- Equation 1: Related to the total number of books.
- Equation 2: Related to the total weight of the books.
Let's denote:
- [tex]\( x \)[/tex] = number of first type books
- [tex]\( y \)[/tex] = number of second type books
The system of equations is:
1) [tex]\( x + y = 179 \)[/tex] (Total number of books)
2) [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex] (Total weight equation)
Thus, the first statement is true: The system of equations is [tex]\( x + y = 179 \)[/tex] and [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex].
2. Check the second statement.
The second statement says the system is [tex]\( x + y = 128 \)[/tex] and [tex]\( \frac{2}{3}x + \frac{3}{4}y = 179 \)[/tex], which is incorrect as it reverses the roles of the total number of books and weight. Hence, the second statement is false.
3. Solve the system of equations.
First, let's clear the fractions in the weight equation.
Multiply the equation [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex] by 12 (common multiple of denominators 3 and 4):
[tex]\[ 8x + 9y = 1536 \][/tex]
Now we solve the system:
1) [tex]\( x + y = 179 \)[/tex]
2) [tex]\( 8x + 9y = 1536 \)[/tex]
We can use the elimination method:
From equation 1, express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 179 - x \][/tex]
Substitute into the second equation:
[tex]\[ 8x + 9(179 - x) = 1536 \][/tex]
Expand and simplify:
[tex]\[ 8x + 1611 - 9x = 1536 \][/tex]
[tex]\[ -x + 1611 = 1536 \][/tex]
[tex]\[ -x = 1536 - 1611 \][/tex]
[tex]\[ -x = -75 \][/tex]
[tex]\[ x = 75 \][/tex]
Substitute [tex]\( x = 75 \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 179 - 75 \][/tex]
[tex]\[ y = 104 \][/tex]
Thus, the solution is [tex]\( x = 75 \)[/tex] and [tex]\( y = 104 \)[/tex].
4. Verify remaining statements:
- Statement 3 (about eliminating [tex]\( x \)[/tex]): To eliminate the [tex]\( x \)[/tex] variable, we actually wouldn't multiply only by 3; rather, we should manipulate the equations as shown above, and multiplying by 3 doesn't directly help eliminate variables. This statement is false.
- Statement 4 (eliminate [tex]\( y \)[/tex]): To eliminate the [tex]\( y \)[/tex] variable, you can indeed multiply the first equation by 3 and adjust the second by another method such that the coefficients of [tex]\( y \)[/tex] become equal, followed by elimination steps similar to above. This statement could be partially true with specific adjustments.
- Statement 5: It's incorrect to say 104 copies are of one type and 24 of the other, because we found 75 copies of the first book and 104 of the second. This statement is false.
Final checked statements are:
- Statement 1: True
- Statement 2: False
- Statement 3: False
- Statement 4: Could be True depending on adjustments
- Statement 5: False
Given:
- 179 copies of books in total.
- Total weight is 128 pounds.
- Each of the first books weighs [tex]\( \frac{2}{3} \)[/tex] pounds.
- Each of the second books weighs [tex]\( \frac{3}{4} \)[/tex] pounds.
We need to verify the given statements.
1. Formulate the system of equations.
The problem gives us two types of equations:
- Equation 1: Related to the total number of books.
- Equation 2: Related to the total weight of the books.
Let's denote:
- [tex]\( x \)[/tex] = number of first type books
- [tex]\( y \)[/tex] = number of second type books
The system of equations is:
1) [tex]\( x + y = 179 \)[/tex] (Total number of books)
2) [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex] (Total weight equation)
Thus, the first statement is true: The system of equations is [tex]\( x + y = 179 \)[/tex] and [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex].
2. Check the second statement.
The second statement says the system is [tex]\( x + y = 128 \)[/tex] and [tex]\( \frac{2}{3}x + \frac{3}{4}y = 179 \)[/tex], which is incorrect as it reverses the roles of the total number of books and weight. Hence, the second statement is false.
3. Solve the system of equations.
First, let's clear the fractions in the weight equation.
Multiply the equation [tex]\( \frac{2}{3}x + \frac{3}{4}y = 128 \)[/tex] by 12 (common multiple of denominators 3 and 4):
[tex]\[ 8x + 9y = 1536 \][/tex]
Now we solve the system:
1) [tex]\( x + y = 179 \)[/tex]
2) [tex]\( 8x + 9y = 1536 \)[/tex]
We can use the elimination method:
From equation 1, express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 179 - x \][/tex]
Substitute into the second equation:
[tex]\[ 8x + 9(179 - x) = 1536 \][/tex]
Expand and simplify:
[tex]\[ 8x + 1611 - 9x = 1536 \][/tex]
[tex]\[ -x + 1611 = 1536 \][/tex]
[tex]\[ -x = 1536 - 1611 \][/tex]
[tex]\[ -x = -75 \][/tex]
[tex]\[ x = 75 \][/tex]
Substitute [tex]\( x = 75 \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 179 - 75 \][/tex]
[tex]\[ y = 104 \][/tex]
Thus, the solution is [tex]\( x = 75 \)[/tex] and [tex]\( y = 104 \)[/tex].
4. Verify remaining statements:
- Statement 3 (about eliminating [tex]\( x \)[/tex]): To eliminate the [tex]\( x \)[/tex] variable, we actually wouldn't multiply only by 3; rather, we should manipulate the equations as shown above, and multiplying by 3 doesn't directly help eliminate variables. This statement is false.
- Statement 4 (eliminate [tex]\( y \)[/tex]): To eliminate the [tex]\( y \)[/tex] variable, you can indeed multiply the first equation by 3 and adjust the second by another method such that the coefficients of [tex]\( y \)[/tex] become equal, followed by elimination steps similar to above. This statement could be partially true with specific adjustments.
- Statement 5: It's incorrect to say 104 copies are of one type and 24 of the other, because we found 75 copies of the first book and 104 of the second. This statement is false.
Final checked statements are:
- Statement 1: True
- Statement 2: False
- Statement 3: False
- Statement 4: Could be True depending on adjustments
- Statement 5: False
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