Question

What value completes the equation of the line that is perpendicular to the line y=−3/4x+7
and goes through the point (6,1)
?(1 point)
The line that is perpendicular to y=−3/4x+7
and passes through the point (6,1)
is y=4/3x+ ____

Answer 1

To find the equation of the line that is perpendicular to \( y = -\frac{3}{4}x + 7 \) and passes through the point \( (6, 1) \), we start by determining the slope of the given line.

The slope of the line \( y = -\frac{3}{4}x + 7 \) is \( -\frac{3}{4} \). The slope of the line that is perpendicular to it will be the negative reciprocal of this slope. The negative reciprocal of \( -\frac{3}{4} \) is:

\[
\text{slope} = \frac{4}{3}
\]

Now that we have the slope of the new line, we can use the point-slope form of a line equation, 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. Here, \( m = \frac{4}{3} \), \( x_1 = 6 \), and \( y_1 = 1 \).

Substituting these values into the equation gives:

\[
y - 1 = \frac{4}{3}(x - 6)
\]

Now, simplify this equation:

\[
y - 1 = \frac{4}{3}x - \frac{24}{3}
\]
\[
y - 1 = \frac{4}{3}x - 8
\]
\[
y = \frac{4}{3}x - 8 + 1
\]
\[
y = \frac{4}{3}x - 7
\]

Thus, the equation of the line that is perpendicular to \( y = -\frac{3}{4}x + 7 \) and goes through the point \( (6,1) \) is:

\[
y = \frac{4}{3}x - 7
\]

Therefore, the value that completes the equation \( y = \frac{4}{3}x + \_\_\_ \) is:

\[
\boxed{-7}
\]