Question

Which lines are perpendicular to the line y – 1 = One-third(x+2)? Check all that apply.

y + 2 = –3(x – 4)
y − 5 = 3(x + 11)
y = -3x – Five-thirds
y = One-thirdx – 2
3x + y = 7

Answers

GPT-4o mini
To determine which lines are perpendicular to the line given by the equation \(y - 1 = \frac{1}{3}(x + 2)\), we first need to convert this equation into slope-intercept form (y = mx + b), where m is the slope.

Starting with:

\[
y - 1 = \frac{1}{3}(x + 2)
\]

Distributing \(\frac{1}{3}\):

\[
y - 1 = \frac{1}{3}x + \frac{2}{3}
\]

Adding 1 to both sides:

\[
y = \frac{1}{3}x + \frac{2}{3} + 1
\]

Simplifying:

\[
y = \frac{1}{3}x + \frac{5}{3}
\]

The slope of this line is \(\frac{1}{3}\). The slopes of perpendicular lines are negative reciprocals, so we need the negative reciprocal of \(\frac{1}{3}\), which is \(-3\).

Now, we will check each given line to see if its slope is \(-3\).

1. **Line: \(y + 2 = -3(x - 4)\)**

Converting this to slope-intercept form:

\[
y + 2 = -3x + 12
\]
\[
y = -3x + 10
\]

Slope = \(-3\) (This is perpendicular)

2. **Line: \(y - 5 = 3(x + 11)\)**

Convert to slope-intercept form:

\[
y - 5 = 3x + 33
\]
\[
y = 3x + 38
\]

Slope = \(3\) (Not perpendicular)

3. **Line: \(y = -3x - \frac{5}{3}\)****

This is already in slope-intercept form.

Slope = \(-3\) (This is perpendicular)

4. **Line: \(y = \frac{1}{3}x - 2\)****

This is already in slope-intercept form.

Slope = \(\frac{1}{3}\) (Not perpendicular)

5. **Line: \(3x + y = 7\)**

Convert to slope-intercept form:

\[
y = -3x + 7
\]

Slope = \(-3\) (This is perpendicular)

In conclusion, the lines that are perpendicular to \(y - 1 = \frac{1}{3}(x + 2)\) are:

- \( y + 2 = -3(x - 4) \)
- \( y = -3x - \frac{5}{3} \)
- \( 3x + y = 7 \)

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