Determine if the given pairs of lines are parallel, perpendicular, or intersecting.
1. \( Y = \frac{7}{3}x - 8 \) and \( Y = \frac{7}{3}x + 3 \)
2. \( Y = 2x - 1 \) and \( x + 2y = 6 \)
3. \( Y = \frac{5}{2}x - 17 \) and \( Y = -\frac{5}{2}x - 1 \)
4. \( Y = \frac{5}{2}x - 4 \) and \( 5x + 2y = 6 \)
5. \( Y = -\frac{2}{3}x + 12 \) and \( Y = \frac{3}{2}x + 3 \)
6. \( Y = \frac{1}{3}x - 13 \) and \( Y = 3x + 2 \)
7. \( Y = -4x + 5 \) and \( x + 4y = -16 \)
8. \( Y = \frac{2}{3}x + 3 \) and \( 2x + 3y = 21 \)
Answer (2)
k : y = m 1 ⋅ x + b 1 an d l : y = m 2 ⋅ x + b 2 k ∣∣ l ⇔ m 1 = m 2 an d k ⊥ l ⇔ m 1 ⋅ m 2 = − 1 − − − − − − − − − − − − − − − − − − − − − − − − − − − − 1. k : y = 3 7 x − 8 an d l : y = 3 7 x + 3 m 1 = m 2 = 3 7 ⇒ k ∣∣ l
2. k : y = 2 x − 1 an d l : − x + 2 y = 6 ⇒ 2 y = x + 6 ⇒ y = 2 1 x + 3 m 1 = m 2 an d m 1 ⋅ m 2 = − 1 ⇒ t h e l in es in t ersec t 3. k : y = 2 5 x − 17 an d l : y = − 2 5 x − 1 ⇒ t h e l in es in t ersec t 4. k : y = 2 5 x − 4 an d l : − 5 x + 2 y = 6 ⇒ 2 y = 5 x + 6 ⇒ y = 2 5 x + 3 m 1 = m 2 = 2 5 ⇒ k ∣∣ l
5. k : y = − 3 2 x + 12 an d l : y = 2 3 x + 3 m 1 ⋅ m 2 = − 3 2 ⋅ 2 3 = − 1 ⇒ k ∣∣ l 6. k : y = 3 1 x − 13 an d l : y = 3 x + 2 ⇒ t h e l in es in t ersec t 7. k : y = − 4 x + 5 , l : − x + 4 y = − 16 ⇒ 4 y = x − 16 ⇒ y = 4 1 x − 4 m 1 ⋅ m 2 = − 4 ⋅ 4 1 = − 1 ⇒ k ⊥ l
8. k : y = 3 2 x + 3 , l : − 2 x + 3 y = 21 ⇒ 3 y = 2 x + 21 ⇒ y = 3 2 x + 7 m 1 = m 2 = 3 2 ⇒ k ∣∣ l
The analysis of the pairs of lines reveals that:
The first pair is parallel.
The second pair is intersecting. The third pair is intersecting, and so on. Other relationships are identified accordingly.
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