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A teacher received the following answer to a quiz question in which the student was asked to simplify the expression in Step 1:

Step 1: [tex] -5(x - 16y) + 7(-2x - 8y) [/tex]
Step 2: [tex] -5x + 80y - 14x - 8y [/tex]
Step 3: [tex] -5x - 14x + 80y - 8y [/tex]
Step 4: [tex] (-14x - 5x) + (80y - 8y) [/tex]
Step 5: [tex] -19x + 72y [/tex]

Part A: Identify the step where the student made a mistake and explain the specific error that was made. Support your answer using the correct vocabulary. (6 points)

Part B: How would you correct the mistake? Show all your work. (6 points)

Sagot :

GinnyAnsweridn
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]
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