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: \[ -3y = 2x + 12 \implies y = -\frac{2}{3}x - 4 \]
     • Slope (m2) = \(-\frac{2}{3}\)

   Since \(m1 = m2\), the lines are parallel.

2. For the second pair of lines:

   • Line 1: \(y - 6 = \frac{1}{4}(x + 2)\) \[ y - 6 = \frac{1}{4}x + \frac{1}{2} \implies y = \frac{1}{4}x + 6.5 \]
     • 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: \(2x - y = 7\) \[ -y = -2x + 7 \implies y = 2x - 7 \]
     • Slope (m1) = \(2\)
   • Line 2: \(3x - 4y = -8\) \[ -4y = -3x - 8 \implies y = \frac{3}{4}x + 2 \]
     • Slope (m2) = \(\frac{3}{4}\)

   The slopes \(m1\) and \(m2\) are not equal and their product is \(2 \cdot \frac{3}{4} \neq -1\). 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