To determine which expressions are equivalent to [tex]\( 8(-10x + 3.5y - 7) \)[/tex], we need to simplify each option and compare it to the given expression.
Let's start with the given expression and each of the options one-by-one.
Given Expression:
[tex]\[ 8(-10x + 3.5y - 7) \][/tex]
First, distribute the 8 in the expression:
[tex]\[ = 8 \cdot (-10x) + 8 \cdot (3.5y) + 8 \cdot (-7) \][/tex]
[tex]\[ = -80x + 28y - 56 \][/tex]
Now, let’s compare this with each provided option.
### Option 1:
[tex]\[ -80x + 24.5y - 58 \][/tex]
This is not equivalent to [tex]\(-80x + 28y - 56\)[/tex] as the coefficients of [tex]\(y\)[/tex] and the constants do not match.
### Option 2:
[tex]\[ -80x + 28y - 58 \][/tex]
This is not equivalent to [tex]\(-80x + 28y - 56\)[/tex] since the constant term does not match.
### Option 3:
[tex]\[ 80x + 28y + 58 \][/tex]
This is clearly not equivalent to [tex]\(-80x + 28y - 56\)[/tex] since both the sign and the constant term are different.
### Option 4:
[tex]\[ 4(-20x + 7y - 14) \][/tex]
First, distribute the 4:
[tex]\[ = 4 \cdot (-20x) + 4 \cdot (7y) + 4 \cdot (-14) \][/tex]
[tex]\[ = -80x + 28y - 56 \][/tex]
This matches the expression [tex]\( -80x + 28y - 56 \)[/tex].
### Option 5:
[tex]\[ -4(-20x + 7y - 14) \][/tex]
First, distribute the -4:
[tex]\[ = -4 \cdot (-20x) + -4 \cdot (7y) + -4 \cdot (-14) \][/tex]
[tex]\[ = 80x - 28y + 56 \][/tex]
This is not equivalent to [tex]\(-80x + 28y - 56\)[/tex] since both the signs of the coefficients and the constant term are reversed.
Given these calculations, the only expression that is equivalent to [tex]\( 8(-10x + 3.5y - 7) \)[/tex] is:
[tex]\[ 4(-20x + 7y - 14) \][/tex]
Therefore, the correct answer is:
[tex]\[ 4(-20x + 7y - 14) \][/tex]
