Are the pairs of lines parallel, perpendicular, or intersecting?(3 points)

Question

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y = −2/3 x + 7
−2x−3y=12

y − 6 = 1/4 (x+2)
y=−4x−10

2x−y=7
3x−4y=−8

Answer 1

To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we need to analyze their slopes.

1. For the first pair of lines:

   \( y = -\frac{2}{3}x + 7 \) has a slope of \( -\frac{2}{3} \).

   The second line is given in standard form \( -2x - 3y = 12 \). We can convert it to slope-intercept form: \[ -3y = 2x + 12 \ y = -\frac{2}{3}x - 4 \] This line also has a slope of \( -\frac{2}{3} \).

   Since both slopes are equal, the lines are parallel.

2. For the second pair of lines:

   The first line is given in point-slope form: \( y - 6 = \frac{1}{4}(x + 2) \), which simplifies to: \[ y = \frac{1}{4}x + 6 + \frac{1}{2} = \frac{1}{4}x + 6.5 \] This line has a slope of \( \frac{1}{4} \).

   The second line \( y = -4x - 10 \) has a slope of \( -4 \).

   The product of the slopes \( \frac{1}{4} \times (-4) = -1 \), which indicates the lines are perpendicular.

3. For the third pair of lines:

   The first line \( 2x - y = 7 \) can be rewritten in slope-intercept form: \[ -y = -2x + 7 \ y = 2x - 7 \] This line has a slope of \( 2 \).

   The second line \( 3x - 4y = -8 \) can also be rewritten: \[ -4y = -3x - 8 \ y = \frac{3}{4}x + 2 \] This line has a slope of \( \frac{3}{4} \).

   These two slopes \( 2 \) and \( \frac{3}{4} \) do not indicate they are equal or negative reciprocals, so the lines are intersecting.

To summarize the relationships:

• First pair: Parallel
• Second pair: Perpendicular
• Third pair: Intersecting