Question

1. lin solved the equation \\(8(x - 3) + 7 = 2x(4 - 17)\\) incorrectly. find the errors in her solution. what should her answer have been?

lins solution:
\\8(x - 3) + 7 = 2x(4 - 17)\\
\\8(x - 3) + 7 = 2x(13)\\
\\8x - 24 + 7 = 26x\\
\\8x - 17 = 26x\\
\\-17 = 34x\\
\\-\frac{1}{2} = x\\

Explanation:

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<reasoning>

Identify errors in Lin's solution

Using the Error Analysis in Equations knowledge point, we examine Lin's steps:

1. In the first step, Lin simplifies \(2x(4 - 17)\) to \(2x(13)\). This is incorrect because \(4 - 17 = -13\), not \(13\).
2. In the fourth step, Lin transitions from \(8x - 17 = 26x\) to \(-17 = 34x\). She added \(8x\) to both sides instead of subtracting \(8x\). It should be \(-17 = 18x\).

Solve the equation correctly

Using the Multi-Step Equations knowledge point, we solve the original equation:
\[
8(x - 3) + 7 = 2x(4 - 17)
\]

Simplify both sides of the equation

First, simplify the terms inside the parentheses on the right side:
\[
8(x - 3) + 7 = 2x(-13)
\]
\[
8(x - 3) + 7 = -26x
\]
Next, apply the distributive property on the left side:
\[
8x - 24 + 7 = -26x
\]
Combine the constant terms on the left side:
\[
8x - 17 = -26x
\]

Isolate the variable x

Subtract \(8x\) from both sides to collect all variable terms on one side:
\[
-17 = -34x
\]

Solve for x

Divide both sides by \(-34\):
\[
x = \frac{-17}{-34} = \frac{1}{2}
\]
</reasoning>

<answer>
Errors in Lin's solution:

1. First Error: In the first step, Lin simplified \(4 - 17\) to \(13\) instead of \(-13\).
2. Second Error: In the fourth step, Lin added \(8x\) to both sides to get \(34x\) instead of subtracting \(8x\) from \(26x\).

Correct Answer:
The correct solution should be \(x = \frac{1}{2}\).
</answer>

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Answer:

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<reasoning>

Identify errors in Lin's solution

Using the Error Analysis in Equations knowledge point, we examine Lin's steps:

1. In the first step, Lin simplifies \(2x(4 - 17)\) to \(2x(13)\). This is incorrect because \(4 - 17 = -13\), not \(13\).
2. In the fourth step, Lin transitions from \(8x - 17 = 26x\) to \(-17 = 34x\). She added \(8x\) to both sides instead of subtracting \(8x\). It should be \(-17 = 18x\).

Solve the equation correctly

Using the Multi-Step Equations knowledge point, we solve the original equation:
\[
8(x - 3) + 7 = 2x(4 - 17)
\]

Simplify both sides of the equation

First, simplify the terms inside the parentheses on the right side:
\[
8(x - 3) + 7 = 2x(-13)
\]
\[
8(x - 3) + 7 = -26x
\]
Next, apply the distributive property on the left side:
\[
8x - 24 + 7 = -26x
\]
Combine the constant terms on the left side:
\[
8x - 17 = -26x
\]

Isolate the variable x

Subtract \(8x\) from both sides to collect all variable terms on one side:
\[
-17 = -34x
\]

Solve for x

Divide both sides by \(-34\):
\[
x = \frac{-17}{-34} = \frac{1}{2}
\]
</reasoning>

<answer>
Errors in Lin's solution:

1. First Error: In the first step, Lin simplified \(4 - 17\) to \(13\) instead of \(-13\).
2. Second Error: In the fourth step, Lin added \(8x\) to both sides to get \(34x\) instead of subtracting \(8x\) from \(26x\).

Correct Answer:
The correct solution should be \(x = \frac{1}{2}\).
</answer>

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