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

the answer choices are
intersecting

perpendicular

parallel

Answer 1

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

1. For the first pair of lines:

   • Line 1: $y=-\frac{2}{3}x+7$
     • Slope (m1) = $-\frac{2}{3}$
   • Line 2: Convert $-2x-3y=12$ to slope-intercept form: $$-3y=2x+12⟹y=-\frac{2}{3}x-4$$
     • Slope (m2) = $-\frac{2}{3}$

   Since $m1=m2$, the lines are parallel.

2. For the second pair of lines:

   • Line 1: $y-6=\frac{1}{4}(x+2)$ $$y-6=\frac{1}{4}x+\frac{1}{2}⟹y=\frac{1}{4}x+6.5$$
     • Slope (m1) = $\frac{1}{4}$
   • Line 2: $y=-4x-10$
     • Slope (m2) = $-4$

   The product of the slopes is $m1\cdot m2=\frac{1}{4}\cdot (-4)=-1$. Thus, the lines are perpendicular.

3. For the third pair of lines:

   • Line 1: $2x-y=7$ $$-y=-2x+7⟹y=2x-7$$
     • Slope (m1) = $2$
   • Line 2: $3x-4y=-8$ $$-4y=-3x-8⟹y=\frac{3}{4}x+2$$
     • Slope (m2) = $\frac{3}{4}$

   The slopes $m1$ and $m2$ are not equal and their product is $2\cdot \frac{3}{4}\ne -1$. Thus, the lines are intersecting.

Combining the results:

1. First pair: Parallel
2. Second pair: Perpendicular
3. Third pair: Intersecting

Responses to input:

• First pair: parallel
• Second pair: perpendicular
• Third pair: intersecting