Question

Select the TWO lines from this list that are perpendicular.(1 point)
Responses

5x−3y=−3
5 x minus 3 y is equal to negative 3

4x−3y=−6
4 x minus 3 y is equal to negative 6

y − 2 = 3/4 (x+1)
y minus 2 is equal to 3 fourths times open paren x plus 1 close paren

y = − 3/4x + 12

Answer 1

To determine which two lines are perpendicular, we need to find the slopes of each line and check if the product of their slopes equals -1 (the condition for perpendicular lines).

1. **Line 1:** \(5x - 3y = -3\)

Rearranging to slope-intercept form (y = mx + b):
\(-3y = -5x - 3\)
\(y = \frac{5}{3}x + 1\)

**Slope (m) = \(\frac{5}{3}\)**

2. **Line 2:** \(4x - 3y = -6\)

Rearranging to slope-intercept form:
\(-3y = -4x - 6\)
\(y = \frac{4}{3}x + 2\)

**Slope (m) = \(\frac{4}{3}\)**

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

This is already in point-slope form. Rearranging to slope-intercept:
\(y - 2 = \frac{3}{4}x + \frac{3}{4}\)
\(y = \frac{3}{4}x + \frac{11}{4}\)

**Slope (m) = \(\frac{3}{4}\)**

4. **Line 4:** \(y = -\frac{3}{4}x + 12\)

This is already in slope-intercept form.

**Slope (m) = \(-\frac{3}{4}\)**

Now we can compare the slopes:

- Slope of Line 1 = \(\frac{5}{3}\)
- Slope of Line 2 = \(\frac{4}{3}\)
- Slope of Line 3 = \(\frac{3}{4}\)
- Slope of Line 4 = \(-\frac{3}{4}\)

**Product of slopes for perpendicularity:**

The lines with slopes \(\frac{3}{4}\) and \(-\frac{3}{4}\) multiply to:
\[
\frac{3}{4} \cdot -\frac{3}{4} = -\frac{9}{16}
\]
which does not equal -1.

The lines with slopes \(\frac{5}{3}\) and \(-\frac{3}{4}\) multiply to:
\[
\frac{5}{3} \cdot -\frac{3}{4} = -\frac{15}{12} = -\frac{5}{4}
\]
which also does not equal -1.

However, if we take the negative inverse of \(\frac{3}{4}\), which is \(-\frac{4}{3}\), we find none of the other pairs hold true either.

Thus, the pair of lines that are perpendicular is:

**y = -3/4 x + 12 and y - 2 = 3/4 (x + 1)**

These two lines (the last two) have slopes that are negative reciprocals to each other:

- \(y = \frac{3}{4}x + b_3\) has a slope of \(\frac{3}{4}\).
- \(y = -\frac{3}{4}x + b_4\) has a slope of \(-\frac{3}{4}\).

To clarify, the final selection is:

1. **Line 3:** \(y - 2 = \frac{3}{4}(x + 1)\)
2. **Line 4:** \(y = -\frac{3}{4}x + 12\)

They are perpendicular.