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Solving Equations with the Variable on Both Sides - Item 3463
Question 4 of 7

Luca shows his work in finding the solution to [tex]$2-6x = 5 - 5(x - 1)$[/tex]. After checking his answer in the original equation, he found that it did not work. Where did he make a mistake?

[tex]\[
\begin{tabular}{|c|c|}
\hline
\textbf{Step} & \textbf{Work} \\
\hline
Step 1 & $2-6x = 5 - 5(x - 1)$ \\
\hline
Step 2 & $2-6x = 5 - 5x + 5$ \\
\hline
Step 3 & $2-6x = 10 - 5x$ \\
\hline
Step 4 & $+5x$ on both sides \\
\hline
Step 5 & $2 - x = 10$ \\
\hline
Step 6 & $-2$ on both sides \\
\hline
Step 7 & $-x = 8$ \\
\hline
Step 8 & $x = -8$ \\
\hline
\end{tabular}
\][/tex]

In which step did Luca make a mistake?



Answer :

GinnyAnswer
To solve the equation [tex]\(2 - 6x = 5 - 5(x - 1)\)[/tex] correctly and identify where Luca made a mistake, let's go through the problem step by step:

Step 1: Start with the given equation:
[tex]\[2 - 6x = 5 - 5(x - 1)\][/tex]

Step 2: Distribute the [tex]\(-5\)[/tex] on the right-hand side:
[tex]\[5 - 5(x - 1) = 5 - 5x + 5\][/tex]
[tex]\[5 - 5x + 5 = 10 - 5x\][/tex]
So the equation becomes:
[tex]\[2 - 6x = 10 - 5x\][/tex]

Step 3: Move all the terms involving [tex]\(x\)[/tex] on one side and constant terms on the other side. Subtract [tex]\(10\)[/tex] from both sides:
[tex]\[2 - 10 - 6x = -5x\][/tex]
[tex]\[-8 - 6x = -5x\][/tex]

Step 4: Simplify the equation by adding [tex]\(6x\)[/tex] to both sides:
[tex]\[-8 - 6x + 6x = -5x + 6x\][/tex]
[tex]\[-8 = x\][/tex]

Step 5: Solve for [tex]\(x\)[/tex]:
[tex]\[x = -8\][/tex]

Thus, Luca's mistake was in the distribution and combination of terms. The correct solution to the equation [tex]\(2 - 6x = 5 - 5(x - 1)\)[/tex] is found by following the correct steps outlined above and solving for [tex]\(x = -8\)[/tex].

Therefore, the correct answer is:
[tex]\[x = -8\][/tex]

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