Question

Which line is perpendicular to the line y=−7/4x
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

Answer 1

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

\[
m = \frac{4}{7}
\]

Now 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, \( (x_1, y_1) = (-7, 5) \) and \( m = \frac{4}{7} \). Substituting in these values, we have:

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

Now, let's simplify the equation:

1. Distribute \( \frac{4}{7} \):

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

2. Add 5 to both sides:

\[
y = \frac{4}{7}x + 4 + 5
\]
\[
y = \frac{4}{7}x + 9
\]

So 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 = \(\frac{4}{7}x + 9\)**.