Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, rewrite the equation in slope intercept form \(\displaystyle (y=mx+b)\):

 \(\displaystyle -9x-9y=9\)

\(\displaystyle -9y=9x+9\)

\(\displaystyle y=-x-1\)

Slope of given line: \(\displaystyle -1\)

Negative reciprocal: \(\displaystyle m=1\)

Rewrite in slope-intercept form:

\(\displaystyle x + y = 17\)

\(\displaystyle x + y -x = 17-x\)

\(\displaystyle y = -x + 17\)

\(\displaystyle y = -1x + 17\)

The slope of the line is the coefficient of \(\displaystyle x\), which is \(\displaystyle -1\). A line perpendicular to this has as its slope the opposite of the reciprocal of \(\displaystyle -1\):

\(\displaystyle m = -\frac{1}{-1} = 1\)

To find the slope of a line perpendicular to a given line, simply take the opposite reciprocal of the slope of the given line.

Since f(x) is given in slope intercept form,

\(\displaystyle y=mx+b \rightarrow m=slope\).

Therefore our original slope is

\(\displaystyle m=\small \frac{5}{6}\)

So our new slope becomes:

\(\displaystyle m_{new}=-\frac{1}{m}=-\frac{1}{\frac{5}{6}}=\small -\frac{6}{5}\)

The equation for a line in standard form is written as follows:

\(\displaystyle y=mx+b\)

Where \(\displaystyle m\) is the slope of the line and \(\displaystyle b\) is the y intercept. By definition, the slope of a line is the negative reciprocal of the slope of the line to which it is perpendicular. So if the given line has a slope of \(\displaystyle -5\), the slope of any line perpendicular to it will have the negative reciprocal of that slope. This gives us:

\(\displaystyle y=-5x-3\rightarrow m=-5\)

\(\displaystyle m_{perp}=-\frac{1}{m}=-\frac{1}{(-5)}=\frac{1}{5}\)

The graph of \(\displaystyle y = b\) for any real number \(\displaystyle b\) is a horizontal line. A line parallel to it is a vertical line, which has a slope that is undefined.

Assume Statement 1 alone. In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. The \(\displaystyle x\)-intercept of the line of the equation can be found by substituting \(\displaystyle y = 0\) and solving for \(\displaystyle x\):

\(\displaystyle 5x+7y = 0\)

\(\displaystyle 5x+7 (0) = 0\)

\(\displaystyle 5x+0 = 0\)

\(\displaystyle 5x=0\)

\(\displaystyle x = 0\)

The \(\displaystyle x\)-intercept of the line is at the origin, \(\displaystyle (0,0)\). It follows that the \(\displaystyle y\)-intercept is also at the origin. Therefore, Statement 1 only gives one point on the line, and its slope cannot be determined.

Assume Statement 2 alone. The slope of the line of the equation \(\displaystyle 5x+7y = 0\) can be calculated by putting it in slope-intercept form \(\displaystyle y = mx+b\):

\(\displaystyle 5x+7y = 0\)

\(\displaystyle 5x+7y -5x = 0 -5x\)

\(\displaystyle 7y = -5x\)

\(\displaystyle 7y\div 7 = -5x \div 7\)

\(\displaystyle y = -\frac{5}{7} x\)

The slope of this line is the coefficient of \(\displaystyle x\), which is \(\displaystyle -\frac{5}{7}\). A line perpendicular to this one has as its slope the opposite of the reciprocal of \(\displaystyle -\frac{5}{7}\), which is

\(\displaystyle - \frac{1}{-\frac{5}{7}} = \frac{7}{5}\).

The question is answered.