In the standard form of a line  the slope is represented by the variable .

In this case the line \(\displaystyle y=5x-32\) has a slope of \(\displaystyle 5\).

The answer is \(\displaystyle 5\).

In the slope-intercept form of a line, , the y-intercept when the line intersects the y-axis.

It does this at \(\displaystyle x=0\).

So we plug \(\displaystyle 0\) in for \(\displaystyle x\) in our equation \(\displaystyle y=4x+15\) to give us \(\displaystyle y=4(0)+15\).

Anything multiplied by \(\displaystyle 0\) is \(\displaystyle 0\) so \(\displaystyle y=15\).

Our coordinates for the y-intercept are \(\displaystyle (0,15)\).

In the standard form of a line  the y-intercept occurs when the line intersects the y-axis.

It does this at 

So we plug  in for  in our equation \(\displaystyle y=9x-17\) to give us \(\displaystyle y=9(0)-17\).

Anything multiplied by  is  so \(\displaystyle -17\).

Our coordinates for the y-intercept are \(\displaystyle (0,-17)\).

Parallel lines have the same slope.

If an equation is in point-slope form, \(\displaystyle y=mx+b\), we take the \(\displaystyle m\) from our equation and set it equal to the slope of our parallel line.

In this case \(\displaystyle m=5\)

The slope of our parallel line is \(\displaystyle 5\).

The point resides in quadrant I (the upper right quadrant), so both values must both be positive. The only possible solution is \(\displaystyle \small (3,5)\).

The slope of a perpendicular lines has the negative reciprocal of the slope of the original line.

If an equation is in point-slope form, , we use the  from our equation as our original slope.

In this case \(\displaystyle m=-15\)

First flip the sign to get \(\displaystyle m=15\). 

To find the reciprocal you take the integer and make it a fraction by placing a \(\displaystyle 1\) over it. If it is already a fraction just flip the numerator and denominator.

Do this to make the slope \(\displaystyle m=\frac{1}{15}\)

The slope of the perpendicular line is \(\displaystyle \frac{1}{15}\).

In the standard form equation of a line, , the slope is represented by the variable .

In this case the line \(\displaystyle y=13x+17\) has a slope of \(\displaystyle 13\).

Therefore the answer is \(\displaystyle m=13\).

In the slope-intercept form of a line, , the slope is represented by the variable .

In this case the line

has a slope of .

The answer is \(\displaystyle m=15\).

In the slope-intercept form of a line, , the y-intercept is when the line intersects the y-axis.

It does this at .

So we plug  in for  in our equation

\(\displaystyle y=31x-65\) 

to give us

\(\displaystyle y=31(0)-65\)

Anything multiplied by  is  so

\(\displaystyle y=-65\)

Our coordinates for the y-intercept are \(\displaystyle (0,-65)\).

Parallel lines have the same slope.

If an equation is in slope-intercept form, , we take the  from our equation and set it equal to the slope of our parallel line.

In this case \(\displaystyle m=16\).

The slope of our parallel line is \(\displaystyle m=16\).