Question

Which line is perpendicular to the line y=−7/4x and goes through the point (−7,5) ?(1 point)
Responses

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

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

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

y=4/7x+9

Answer 1

To find a line that is perpendicular to the line given by \( y = -\frac{7}{4}x \) and goes through the point \((-7, 5)\), we first need to determine the slope of the perpendicular line.

The slope of the line \( y = -\frac{7}{4}x \) is \( -\frac{7}{4} \). The slope of a line that is perpendicular to this will be the negative reciprocal of \( -\frac{7}{4} \).

The negative reciprocal of \( -\frac{7}{4} \) is:

\[
\text{slope} = \frac{4}{7}
\]

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

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

where \( (x_1, y_1) \) is the point through which the line passes and \( m \) is the slope. In this case, \( (x_1, y_1) = (-7, 5) \) and \( m = \frac{4}{7} \).

Substituting these values into the point-slope formula gives:

\[
y - 5 = \frac{4}{7}(x + 7)
\]

Now we can simplify this equation to slope-intercept form \( y = mx + b \):

1. Distribute \( \frac{4}{7} \) on the right side:

\[
y - 5 = \frac{4}{7}x + \frac{4}{7} \cdot 7
\]

\[
y - 5 = \frac{4}{7}x + 4
\]

2. Add 5 to both sides:

\[
y = \frac{4}{7}x + 4 + 5
\]

\[
y = \frac{4}{7}x + 9
\]

This means the equation of the line that is perpendicular to \( y = -\frac{7}{4}x \) and passes through the point \((-7, 5)\) is:

\[
y = \frac{4}{7}x + 9
\]

Thus, the correct response is:

**y = 4/7x + 9**