Question

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

y−2=34(x+1)
y minus 2 is equal to 3 fourths times open paren x plus 1 close paren

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=−34x+12

Answer 1

To determine which two lines are perpendicular, we need to find the slopes of the lines and see if they are negative reciprocals of each other.

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

To find the slope from this equation, we can rewrite it in slope-intercept form (y = mx + b):

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

This simplifies to:

\( y = \frac{3}{4}x + \frac{11}{4} \)

The slope (m) of Line 1 is \( \frac{3}{4} \).

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

We can also put this line into slope-intercept form:

\( -3y = -5x - 3 \)

Dividing the entire equation by -3 gives:

\( y = \frac{5}{3}x + 1 \)

The slope (m) of Line 2 is \( \frac{5}{3} \).

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

Putting this into slope-intercept form:

\( -3y = -4x - 6 \)

Dividing through by -3:

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

The slope (m) of Line 3 is \( \frac{4}{3} \).

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

This is already in slope-intercept form, so the slope (m) is \( -\frac{3}{4} \).

Now, we check the slopes. Lines are perpendicular if the product of their slopes is -1.

- **Between Line 1 and Line 4:**
- Slope of Line 1: \( \frac{3}{4} \)
- Slope of Line 4: \( -\frac{3}{4} \)
- Product: \( \frac{3}{4} \times -\frac{3}{4} = -\frac{9}{16} \) (not perpendicular)

- **Between Line 2 and Line 3:**
- Slope of Line 2: \( \frac{5}{3} \)
- Slope of Line 3: \( \frac{4}{3} \)
- Product: \( \frac{5}{3} \times \frac{4}{3} = \frac{20}{9} \) (not perpendicular)

- **Between Line 1 and Line 2:**
- Slope of Line 1: \( \frac{3}{4} \)
- Slope of Line 2: \( \frac{5}{3} \)
- Product: \( \frac{3}{4} \times \frac{5}{3} = \frac{15}{12} = \frac{5}{4} \) (not perpendicular)

- **Between Line 1 and Line 3:**
- Slope of Line 1: \( \frac{3}{4} \)
- Slope of Line 3: \( \frac{4}{3} \)
- Product: \( \frac{3}{4} \times \frac{4}{3} = 1 \) (not perpendicular)

From the examination, **Line 1** (\(y - 2 = \frac{3}{4}(x + 1)\)) and **Line 4** (\(y = -\frac{3}{4}x + 12\)) are the only lines that have slopes that multiply to -1. So, the perpendicular lines are:

**Line 1 and Line 4.**