Certainly! Let's take a detailed look at the steps provided and identify where the student made a mistake.
Part A: Identifying the Mistake
Step 1: [tex]\[-5(x - 16 y) + 7(-2 x - 8 y)\][/tex]
Step 2: [tex]\[-5 x + 80 y - 14 x - 8 y\][/tex]
Here is where we need to be careful. Expand both terms by distributing the constants:
[tex]\[ -5(x - 16y) \Rightarrow -5 \cdot x + (-5) \cdot (-16y) = -5x + 80y \][/tex]
[tex]\[ 7(-2x - 8y) \Rightarrow 7 \cdot (-2x) + 7 \cdot (-8y) = -14x - 56y \][/tex]
So, the correct expansion should be:
[tex]\[ -5 x + 80 y - 14 x - 56 y \][/tex]
The student wrote: [tex]\[-5 x + 80 y - 14 x - 8 y\][/tex]
The mistake here was in simplifying [tex]\(7(-8 y)\)[/tex] as [tex]\(-8 y\)[/tex] instead of the correct [tex]\(-56 y\)[/tex].
Step 3: [tex]\[-5 x - 14 x + 80 y - 8 y\][/tex]
Step 4: [tex]\[(-14 x - 5 x) + (80 y - 8 y)\][/tex]
Step 5: [tex]\[-19 x + 72 y\][/tex]
The error in Step 2 propagated through the remaining steps.
Thus, the mistake is in Step 2 where the incorrect term [tex]\(-8 y\)[/tex] was written instead of [tex]\(-56 y\)[/tex].
Part B: Correcting the Mistake
Let's correct the mistake and show all work step-by-step.
Step 1: [tex]\[-5(x - 16 y) + 7(-2 x - 8 y)\][/tex]
Step 2: Expand both terms correctly:
[tex]\[ -5(x - 16 y) + 7(-2 x - 8 y) \Rightarrow -5 x + 80 y - 14 x - 56 y \][/tex]
Combining like terms:
[tex]\[ -5 x - 14 x + 80 y - 56 y \][/tex]
Step 3: Combine the x-terms and y-terms separately:
[tex]\[ (-5 x - 14 x) + (80 y - 56 y) \][/tex]
Step 4: Simplify each group:
[tex]\[ -19 x + 24 y\][/tex]
Thus, the corrected expression is:
[tex]\[ -19 x + 24 y\][/tex]
