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

Question

Responses

y=47x+9
y equals Start Fraction 4 over 7 End Fraction x plus 9

y=−47x+1
y equals negative Start Fraction 4 over 7 End Fraction x plus 1

y=47x+5
y equals Start Fraction 4 over 7 End Fraction x plus 5

y=−74x+5
y equals negative Start Fraction 7 over 4 End Fraction x plus 5

Answer 1

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

1. Find the slope of the original line: The line \( y = -74x \) has a slope of \(-74\).

2. Calculate the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \(-74\) is: \[ \text{slope} = \frac{1}{74} \]

3. Use the point-slope form of a line: We use the point-slope form of a line, which is: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through. Plugging in our values: \[ y - 5 = \frac{1}{74}(x + 7) \]

4. Rearranging into slope-intercept form: \[ y - 5 = \frac{1}{74}x + \frac{7}{74} \] \[ y = \frac{1}{74}x + 5 + \frac{7}{74} \] To combine \(5\) and \(\frac{7}{74}\), we rewrite \(5\) as \(\frac{370}{74}\): \[ y = \frac{1}{74}x + \frac{370 + 7}{74} = \frac{1}{74}x + \frac{377}{74} \]

5. Finding the equivalent line form: The line can also be expressed as \( y = \frac{1}{74}x + \frac{377}{74} \). However, let's check if any given options match, and note how they transform.

Checking the provided options:

Given:

• \( y = 47x + 9 \)
• \( y = -47x + 1 \)
• \( y = 47x + 5 \)
• \( y = -74x + 5 \)

Reviewing the choices for slopes:

• The slopes \(47\), \(-47\), and \(-74\) do not yield a perpendicular match with the negative reciprocal of \(-74\) which we calculated.

None of the options correspond directly to our derived slope of \(\frac{1}{74}\). It appears there may have been an expectation for an adjusted format, but to achieve perpendicularity from the equation dictated, sticking to derived values went awry in matching options.

Thus, we can't find an available answer from those provided matching the criteria for perpendicular through (-7,5) with accurate slope of \(\frac{1}{74}\) leading toward adjustment or fractional inconsistency with values near or around the input evaluations.

However, it appears the correct slope switches based on the descriptive derivation through the angles, first verification with interactions implies insight into possible alternate degree conditions junctioned. Let's refer back confirming individually within contacts to ensure lining consistency across partial distinctions \(y=\frac{1}{74}x,\).

Final Answer:

None of the available options fit these criteria exactly.