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Calculate the equation for a line that is perpendicular to the graphed line.

(1 point)
Responses

y=32x+6y is equal to 3 halves x plus 6

y=−23x+4y is equal to negative 2 thirds x plus 4

y=23x+6y is equal to 2 thirds x plus 6

y=32x+4

1 answer

To find the equation of a line that is perpendicular to a given line, you need to know the slope of the original line. The slope of the perpendicular line will be the negative reciprocal of the slope of the original line.

From the responses you've provided, we first need to identify the slopes:

  1. y = 32x + 6: The slope is 32.
  2. y = −23x + 4: The slope is -23.
  3. y = 23x + 6: The slope is 23.
  4. y = 32x + 4: The slope is 32.

Now, we find the negative reciprocals of these slopes:

  • For 32, the negative reciprocal is \(-\frac{1}{32}\).
  • For -23, the negative reciprocal is \(\frac{1}{23}\).
  • For 23, the negative reciprocal is \(-\frac{1}{23}\).

Here are the negative reciprocals and the corresponding perpendicular lines:

  1. For y = 32x + 6: Perpendicular slope = \(-\frac{1}{32}\)
  2. For y = −23x + 4: Perpendicular slope = \(\frac{1}{23}\)
  3. For y = 23x + 6: Perpendicular slope = \(-\frac{1}{23}\)
  4. For y = 32x + 4: Perpendicular slope = \(-\frac{1}{32}\)

Now, we check which of the provided options has the corresponding negative reciprocal slope (and can be simplified into the correct form if necessary). The options given appear more like responses for different equations rather than the slopes of perpendicular lines. However, we can determine that only the equations with slopes of negative reciprocals are valid answers if we match their slopes based on the given information.

  1. y = 32x + 6 (slope = 32) => Perpendicular: \(y = -\frac{1}{32}x + b\)
  2. y = −23x + 4 (slope = -23) => Perpendicular: \(y = \frac{1}{23}x + b\)
  3. y = 23x + 6 (slope = 23) => Perpendicular: \(y = -\frac{1}{23}x + b\)
  4. y = 32x + 4 (slope = 32) => Perpendicular: \(y = -\frac{1}{32}x + b\)

Since all we can see from your options are the same slopes repeated in different ways, we need to ensure that there might be an inconsistency. If the original line confirms to be one of these options, pick the negative reciprocal of its slope based on the rules of perpendicularity.

If you can identify which line these options correspond to, please let me know, and I can help you derive the specific equation for the perpendicular line.