Problem Analysis:
The task involves simplifying the expression \((10x)(8) - 6(x - 3)\). Four individuals—Karen, Lucy, Jack, and Marcus—have attempted to simplify this expression, but their methods and results differ. The goal is to identify the correct simplification process and determine which individual(s) arrived at the correct answer.
Expression to Simplify:
\[
(10x)(8) - 6(x - 3)
\]
Step-by-Step Simplification:
1.
Simplify \((10x)(8)\):
\[
(10x)(8) = 80x
\]
So the expression becomes:
\[
80x - 6(x - 3)
\]
2.
Distribute \(-6\) across \((x - 3)\):
\[
-6(x - 3) = -6x + 18
\]
So the expression now is:
\[
80x - 6x + 18
\]
3.
Combine like terms:
\[
80x - 6x = 74x
\]
Therefore, the simplified expression is:
\[
74x + 18
\]
Evaluation of Each Individual's Work:
#### Karen:
\[
\begin{aligned}
&(10x)(8) - 6(x - 3) \\
&10x + 2(x - 3) \\
&10x + 2x - 6 \\
&12x - 6
\end{aligned}
\]
-
Error: Karen incorrectly simplified \((10x)(8)\) to \(10x\) instead of \(80x\). This is a fundamental mistake in the first step.
-
Result: \(12x - 6\) (incorrect).
#### Lucy:
\[
\begin{aligned}
&(10x)(8) - 6(x - 3) \\
&80x - 6(-2x) \\
&80x + 12x \\
&92x
\end{aligned}
\]
-
Error: Lucy incorrectly distributed \(-6\) as \(-6(-2x)\) instead of \(-6(x - 3)\). This leads to an incorrect term \(+12x\) instead of \(-6x + 18\).
-
Result: \(92x\) (incorrect).
#### Jack:
\[
\begin{aligned}
&(10x)(8) - 6(x - 3) \\
&80x - 6(x - 3) \\
&80x - 6x + 18 \\
&74x + 18
\end{aligned}
\]
-
Correct Steps: Jack correctly simplified \((10x)(8)\) to \(80x\), distributed \(-6\) as \(-6x + 18\), and combined like terms to get \(74x + 18\).
-
Result: \(74x + 18\) (correct).
#### Marcus:
\[
\begin{aligned}
&(10x)(8) - 6(x - 3) \\
&80x - 6(x - 3) \\
&80x - 6x - 18 \\
&74x - 18
\end{aligned}
\]
-
Error: Marcus incorrectly distributed \(-6\) as \(-6x - 18\) instead of \(-6x + 18\). This leads to an incorrect constant term \(-18\) instead of \(+18\).
-
Result: \(74x - 18\) (incorrect).
Correct Answer:
The correct simplification of the expression \((10x)(8) - 6(x - 3)\) is:
\[
\boxed{74x + 18}
\]
Final Answer:
\[
\boxed{Jack}
\]
Parent Tip: Review the logic above to help your child master the concept of algebraic expressions 7th grade.
