Select the pair of equations whose graphs are perpendicular.
A. \(2y = -3x + 5\) and \(2x + 3y = 4\)
B. \(5x - 8y = 9\) and \(12x - 5y = 7\)
C. \(y = 2x - 7\) and \(x + 2y = 3\)
D. \(x + 6y = 8\) and \(y = 2x - 8\)
Answer (3)
k : y = m 1 x + b 1 an d l : y = m 2 x + b 2 t h e l in e \k i s p er p e n d i c u l a r t o t h e l in e l ⇔ m 1 ⋅ m 2 = − 1 − − − − − − − − − − − − − − − − − − − − − − − − − − − − A . 2 y = − 3 x + 5 ⇒ y = − 2 3 x + 2.5 ⇒ m 1 = − 2 3 2 x + 3 y = 4 ⇒ 3 y = − 2 x + 4 ⇒ y = − 3 2 x + 1 3 1 ⇒ m 2 = − 3 2 m 1 ⋅ m 2 = − 2 3 ⋅ ( − 3 2 ) = 1 = − 1
B . 5 x − 8 y = 9 ⇒ − 8 y = − 5 x + 9 ⇒ y = 8 5 x − 1 8 1 ⇒ m 1 = 8 5 12 x − 5 y = 7 ⇒ − 5 y = − 12 x + 7 ⇒ y = 5 12 x − 1 5 2 ⇒ m 2 = 5 12 m 1 ⋅ m 2 = 8 5 ⋅ 5 12 = 2 3 = − 1
C . y = 2 x − 7 ⇒ m 1 = 2 x + 2 y = 3 ⇒ 2 y = − x + 3 ⇒ y = − 2 1 x + 1 2 1 ⇒ m 2 = − 2 1 m 1 ⋅ m 2 = 2 ⋅ ( − 2 1 ) = − 1 ⇒ y = 2 x − 7 ⊥ x + 2 y = 3
D . x + 6 y = 8 ⇒ 6 y = − x + 8 ⇒ y = − 6 1 x + 1 3 1 ⇒ m 1 = − 6 1 y = 2 x − 8 ⇒ m 2 = 2 m 1 ⋅ m 2 = − 6 1 ⋅ 2 = − 3 1 = − 1
To determine which pair of equations represent perpendicular lines, we need to find the slopes of the lines represented by each equation. Two lines are perpendicular if the product of their slopes is -1, which means their slopes are negative reciprocals of each other. For linear equations in the form y = mx + b, m is the slope.
Let's look at each pair of equations and rewrite them in slope-intercept form if necessary:
A. 2y = –3x + 5 becomes y = –3/2x + 5/2 with a slope of –3/2. 2x + 3y = 4 becomes y = –2/3x + 4/3 with a slope of –2/3.
B. 5x – 8y = 9 becomes y = 5/8x – 9/8 with a slope of 5/8. 12x – 5y = 7 becomes y = 12/5x – 7/5 with a slope of 12/5.
C. y = 2x – 7 with a slope of 2. x + 2y = 3 becomes y = –1/2x + 3/2 with a slope of –1/2.
D. x + 6y = 8 becomes y = –1/6x + 4/3 with a slope of –1/6. y = 2x – 8 with a slope of 2.
Now, by comparing slopes, we see that the correct answer is option C. The slopes 2 and –1/2 are negative reciprocals of each other, which means the lines are perpendicular.
The pair of equations whose graphs are perpendicular is option C: y = 2 x − 7 and x + 2 y = 3 , as their slopes multiply to -1.
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