Question

Parallel & Perpendicular Lines Quick Check
4 of 54 of 5 Items
Question
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=−4/7x+1
y equals negative Start Fraction 4 over 7 End Fraction x plus 1

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

y=4/7x+5

Answer 1

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

1. **Identify the slope of the given line**:
The slope of the line \( y = -\frac{7}{4}x \) is \( -\frac{7}{4} \).

2. **Determine the slope of the perpendicular line**:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of \(-\frac{7}{4}\) is:
\[
\frac{4}{7}
\]

3. **Use the point-slope form to find the equation of the perpendicular line**:
Using the point \((-7, 5)\) and the slope \(\frac{4}{7}\), we can use the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( (x_1, y_1) = (-7, 5) \) and \( m = \frac{4}{7} \):
\[
y - 5 = \frac{4}{7}(x + 7)
\]

To write it in slope-intercept form (y = mx + b), we can expand and simplify:
\[
y - 5 = \frac{4}{7}x + 4
\]
\[
y = \frac{4}{7}x + 9
\]

Now we look at the provided responses:

1. \( y = \frac{4}{7}x + 9 \)
2. \( y = -\frac{4}{7}x + 1 \)
3. \( y = -\frac{7}{4}x + 5 \)
4. \( y = \frac{4}{7}x + 5 \)

The first option, \( y = \frac{4}{7}x + 9 \), is the correct line that is perpendicular to \( y = -\frac{7}{4}x \) and passes through the point \((-7, 5)\).

Thus, the answer is:
**y = \frac{4}{7}x + 9**.