To solve the problem, let's carefully and correctly expand the given expression step-by-step and identify the errors made by Daniel.
The original expression is:
[tex]$
-2 \left( -8x - 4y + \frac{3}{4} \right)
$[/tex]
1. First Step: Distribute the [tex]\(-2\)[/tex] across each term inside the parentheses.
### Distributing [tex]\(-2\)[/tex] to [tex]\(-8x\)[/tex]:
[tex]$
-2 \times -8x = 16x
$[/tex]
The term is positive because the product of two negative numbers is positive.
### Distributing [tex]\(-2\)[/tex] to [tex]\(-4y\)[/tex]:
[tex]$
-2 \times -4y = 8y
$[/tex]
Again, this term is positive for the same reason.
### Distributing [tex]\(-2\)[/tex] to [tex]\(\frac{3}{4}\)[/tex]:
[tex]$
-2 \times \frac{3}{4} = -\frac{6}{4} = -1.5 \quad \text{(or equivalently, } -1 \frac{1}{2}\text{)}
$[/tex]
This term remains negative since multiplying a negative number by a positive number results in a negative number.
So, the expanded form of the original expression should be:
[tex]$
16x + 8y - 1.5 \quad \text{(or equivalently, } 16x + 8y - 1 \frac{1}{2}\text{)}
$[/tex]
### Comparing with Daniel's Result:
Daniel's result was:
[tex]$
10x - 8y - 1 \frac{1}{4}
$[/tex]
### Errors Daniel Made:
1. Term Error in [tex]\(x\)[/tex]-Coefficient:
- The first term should be [tex]\(16x\)[/tex], not [tex]\(10x\)[/tex]. He incorrectly multiplied the coefficients, which should be checked as outlined above.
2. Sign Error in [tex]\(y\)[/tex]-Coefficient:
- The second term should be [tex]\(8y\)[/tex], not [tex]\(-8y\)[/tex]. He did not correctly handle the sign change when distributing the [tex]\(-2\)[/tex] to [tex]\(-4y\)[/tex].
3. Fraction Error in the Constant Term:
- The constant term should be [tex]\( -1.5 \)[/tex] [tex]\((or \, -1 \frac{1}{2})\)[/tex], not [tex]\( -1.25 \)[/tex] [tex]\((or \, -1 \frac{1}{4}) \)[/tex].
Hence, the errors he made are:
- The first term should be negative.
- The second term should be positive.
- The last term should be [tex]\( -1 \frac{1}{2} \)[/tex], not [tex]\( -1 \frac{1}{4} \)[/tex].
These errors are aligned with options presented:
- The first term should be negative.
- The second term should be positive.
- The last term should be [tex]\( -1 \frac{1}{2} \)[/tex], not [tex]\( -1 \frac{1}{4} \)[/tex].
