Suggested review of lines from the
Algebra I curriculum:
Recall:
$$\text{slope of line through points } (x_1,y_1) \text{ and } (x_2,y_2) \ \ =\ \ m \ \ =\ \ \frac{\text{rise}}{\text{run}} \ \ =\ \ \frac{\text{change in } y}{\text{change in } x}\ \ =\ \ \frac{\Delta y}{\Delta x} \ \ =\ \ \frac{y_2  y_1}{x_2  x_1}$$
A common scenario is (see diagram at right):
 you have a line with known slope $\,m\,$
 you know the coordinates of one point on the line;
call this known point $(x_{\text{old}},y_{\text{old}})$
 there is another point on the line whose coordinates are needed;
call this desired point $(x_{\text{new}},y_{\text{new}})$
 you know the change in $\,x\,$ between the old and new point:
$$
\Delta x = x_{\text{new}}  x_{\text{old}}
$$
$$
\begin{alignat}{2}
\Delta x > 0\,\ \ &\iff\ \ &&\text{the new point lies to the right of the old point}\cr
\Delta x < 0\,\ \ &\iff\ \ &&\text{the new point lies to the left of the old point}
\end{alignat}
$$
 you want the $y$coordinate, $\,y_{\text{new}}\,$, of the new point


Solving for $\,y_{\text{new}}\,$ in terms of the known quantities:
$\displaystyle
m = \frac{y_{\text{new}}  y_{\text{old}}}{x_{\text{new}}  x_{\text{old}}}$
$\Rightarrow\ \ y_{\text{new}}  y_{\text{old}} = m
\overbrace{(x_{\text{new}}  x_{\text{old}})}^{\Delta x}$
$\Rightarrow\ \ y_{\text{new}} = y_{\text{old}} + m\Delta x$
EXAMPLE:
You have a known point $\,(1,5)\,$ on a line with slope $\,7.5\,$.
When $\,x = 1.4\,$, what is the $y$value of the point on the line?
SOLUTION:
The change in $\,x\,$ in going from the known point ($x = 1$) to the new point ($x = 1.4$) is: $\,\Delta x = 1.4  1 = 0.4\,$
$y_{\text{new}} = y_{\text{old}} + m\Delta x = 5 + (7.5)(0.4) = 2$