Exercise 2 Page 175

The formula for slope can be written as:

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}},

where

m

is the slope and

(x_{1}, y_{1})

and

(x_{2}, y_{2})

are two arbitrary points on the line. We have been given a slope of

4

and a point on the line,

(3, - 2) .

If we substitute this information into the Slope Formula, we have:

4 = \frac{y _{2} - ( - 2 )}{x _{2} - 3} .

We can simplify this equation to be in the point-slope form,

y - y_{1} = m (x - x_{1}) .

First, we can remove the subscripts because there is only one of each variable.

4 = \frac{y _{2} - ( - 2 )}{x _{2} - 3}

Rewrite $y_{2}$ as $y$

4 = \frac{y - ( - 2 )}{x _{2} - 3}

Rewrite $x_{2}$ as $x$

4 = \frac{y - ( - 2 )}{x - 3}

Next we can simplify and use inverse operations to rearrange the equation to be in the correct form.

4 = \frac{y - ( - 2 )}{x - 3}

$a - (- b) = a + b$

4 = \frac{y + 2}{x - 3}

$LHS \cdot (x - 3) = RHS \cdot (x - 3)$

4 (x - 3) = y + 2

Rearrange equation

y + 2 = 4 (x - 3)

As we have shown, the Slope Formula and the point-slope form are very closely related. We can easily find one when given the other.