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Which line is perpendicular to the line y=−74x and goes through the point (−7,5) ?(1 point)

Responses

y=4/7x+9
y equals Start Fraction 4 over 7 End Fraction x plus 9

y=−7/4x+5
y equals negative Start Fraction 7 over 4 End Fraction x plus 5

y=4/7x+5
y equals Start Fraction 4 over 7 End Fraction x plus 5

y=−4/7x+1

1 answer

To find a line that is perpendicular to the line \( y = -74x \) and passes through the point \((-7, 5)\), we first need to determine the slope of the given line.

The slope of the line \( y = -74x \) is \(-74\). The slope of a line that is perpendicular to another is the negative reciprocal of that slope. Therefore, the slope of the line perpendicular to \( y = -74x \) is:

\[ m = -\frac{1}{-74} = \frac{1}{74} \]

Next, we can use the point-slope form of the equation of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

Here, \( m \) is the slope we just calculated, and \((x_1, y_1) = (-7, 5)\):

Substituting in the values:

\[ y - 5 = \frac{1}{74}(x + 7) \]

Now, to write that in slope-intercept form (\( y = mx + b \)), we will simplify it:

\[ y - 5 = \frac{1}{74}x + \frac{7}{74} \] \[ y = \frac{1}{74}x + 5 + \frac{7}{74} \] \[ y = \frac{1}{74}x + \frac{370 + 7}{74} \quad (\text{since } 370 = 5 \times 74) \] \[ y = \frac{1}{74}x + \frac{377}{74} \]

Now looking at the given options, none of them directly match our result, but we can tell that there is a misinterpretation with the choices provided in the responses.

Let's check through the answers:

  1. y = \(\frac{4}{7}x + 9\) - This has a different slope, so it does not apply.
  2. y = \(-\frac{7}{4}x + 5\) - This also has a different slope.
  3. y = \(\frac{4}{7}x + 5\) - Again, a different slope.
  4. y = \(-\frac{4}{7}x + 1\) - This also does not match.

It seems there might be an error in the options you provided. If \( -74 \) was misinterpreted and you meant a different slope, we could double-check the slopes or confirm whether any options are appropriate.

However, using the negative reciprocal slope of \(\frac{1}{74}\) suggests that none of the provided equations are correct for the conditions stated in your question.

If you are looking to select an answer despite this fact, the slope that is closest to positive \( m \) might appear as \( y = \frac{4}{7}x + 5 \), but it’s not perpendicular to the original line. Ensure the setup matching the true calculations for precise correctness.