To solve the problem, we need to simplify the given fraction [tex]\(\frac{-50 z^2 + 35 z}{-45 z^2 + 40 z}\)[/tex]. Let's break it down step-by-step:
1. Factor the numerator and the denominator:
The given fraction is [tex]\(\frac{-50 z^2 + 35 z}{-45 z^2 + 40 z}\)[/tex].
- First, factor out the greatest common factor (GCF) from the numerator [tex]\(-50 z^2 + 35 z\)[/tex]:
[tex]\[
-50 z^2 + 35 z = 5z (-10z + 7)
\][/tex]
- Next, factor out the greatest common factor (GCF) from the denominator [tex]\(-45 z^2 + 40 z\)[/tex]:
[tex]\[
-45 z^2 + 40 z = 5z (-9z + 8)
\][/tex]
So, the expression can be rewritten using these factorizations:
[tex]\[
\frac{-50 z^2 + 35 z}{-45 z^2 + 40 z} = \frac{5z (-10z + 7)}{5z (-9z + 8)}
\][/tex]
2. Simplify the fraction:
Since [tex]\(5z\)[/tex] is a common factor in both the numerator and the denominator, we can cancel it out (provided [tex]\(z \neq 0\)[/tex]):
[tex]\[
\frac{5z (-10z + 7)}{5z (-9z + 8)} = \frac{-10z + 7}{-9z + 8}
\][/tex]
3. Conclusion:
The simplified expression is [tex]\(\frac{-10z + 7}{-9z + 8}\)[/tex].
Therefore, the expression that is equivalent to [tex]\(\frac{-50 z^2 + 35 z}{-45 z^2 + 40 z}\)[/tex] is given by answer choice:
(A) [tex]\(\frac{-10 z+7}{-9 z+8}\)[/tex]
