Step 1: Name the slope of the line you were given.

The equation $y = - 2x + 3$ is expressed in slope-intercept form.

The slope is $m = - 2$.

Step 2: Determine the slope of the line you need to write an equation for.

You need to write an equation of a line that is parallel to $y = - 2x + 3$.

The new equation must have a slope of $m = - 2$ because parallel lines have identical slope values.

Step 3: Use the slope and the information in the problem to write an equation in point-slope form.

The slope of the parallel line will be $m = - 2$. The ordered pair the parallel line passes through is $( - 4, - 1)$.

$y - \left( - 1 \right) = - 2(x - \left( - 4 \right))$

Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary.

$y - \left( - 1 \right) = - 2\left( x - \left( - 4 \right) \right)$

$y + 1 = - 2\left( x + 4 \right)$

$y + 1 = - 2x - 8$

$y = - 2x - 9$

Step 1: Name the slope of the line you were given.

The equation $y = - \frac{4}{3}x + 20$ is expressed in slope-intercept form.

The slope is $m = - \frac{4}{3}$.

Step 2: Determine the slope of the line you need to write an equation for.

You need to write an equation of a line that has the same steepness as the original. This means the second line needs to have the same slope as the first.

The new equation must have a slope of $m = - \frac{4}{3}$.

Step 3: Use the slope and the information in the problem to write an equation in point-slope form.

The slope of the parallel line will be $m = - \frac{4}{3}$. The ordered pair the parallel line passes through is $(0,\ 12)$.

$y - 12 = - \frac{4}{3}(x - 0)$

Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary.

$y - 12 = - \frac{4}{3}\left( x - 0 \right)$

$y - 12 = - \frac{4}{3}x$

$y = - \frac{4}{3}x + 12$

Note: You could also recognize the given point, $\left( 0,\ 12 \right)$, is the $y$-intercept of the parallel line.

Step 1: Name the slope of the line you were given.

The equation $y + 1 = \frac{4}{9}(x - 18)$ is expressed in point-slope form.

The slope is $m = \frac{4}{9}$.

Step 2: Determine the slope of the line you need to write an equation for.

You need to write an equation of a line that is perpendicular to this one.

Non-vertical perpendicular lines have opposite reciprocal slopes. The new equation must have a slope of $m = - \frac{9}{4}$.

Step 3: Use the slope and the information in the problem to write an equation in point-slope form.

The slope of the perpendicular line will be $m = - \frac{9}{4}$. The ordered pair the perpendicular line passes through is $(3,\ 9)$.

$y - 9 = - \frac{9}{4}(x - 3)$

Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary.

The instructions indicate to express the final equation in point-slope form. No rewriting is necessary.

Step 1: Name the slope of the line you were given.

The slope of the equation $y - 4 = - \frac{5}{3}(x - 3)$ is $m = - \frac{5}{3}$.

Step 2: Determine the slope of the line you need to write an equation for.

Non-vertical perpendicular lines have opposite reciprocal slopes. The new equation must have a slope of $m = \frac{3}{5}$.

Step 3: Use the slope and the information in the problem to write an equation in point-slope form.

The slope of the perpendicular line will be $m = \frac{3}{5}$. The ordered pair the perpendicular line passes through is $( - 16,\ 14)$.

$y - 14 = \frac{3}{5}(x - ( - 16))$

Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary.

$y - 14 = \frac{3}{5}\left( x - \left( - 16 \right) \right)$

$y - 14 = \frac{3}{5}\left( x + 16 \right)$

$y - 14 = \frac{3}{5}x + 9.6$

$y = \frac{3}{5}x + 23.6$

Step 1: Name the slope of the line you were given.

The equation of the $x$-axis is $y = 0$. The slope of this line is $m = 0$.

Step 2: Determine the slope of the line you need to write an equation for.

Parallel lines have identical slope.

The new equation must have a slope of $m = 0$.

Step 3: Use the slope and the information in the problem to write an equation in point-slope form.

The slope of the parallel line will be $m = 0$. The ordered pair the parallel line passes through is $(0,\ 2)$.

$y - 2 = 0(x - 0)$

Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary.

The instructions indicate to express the final equation in point-slope form. No rewriting is necessary.