POINT-SLOPE FORM

Suppose a line has slope $\,m\,$ and passes through the point $\,(x_1,y_1)\,$.
That is, we know the slope of the line and we know a point on the line.
We can get an equation that is ideally suited to these two pieces of information, as follows:

Let $\,(x,y)\,$ denote any other point on the line.
Now, we have two points:   the known point $\,(x_1,y_1)\,$ and a ‘generic’ point $\,(x,y)\,$.
The slope of the line, computed using these two points, must equal $\,m\,$.
Using the slope formula, we have:

$\displaystyle\,m = \frac{y-y_1}{x-x_1}\,$
or, equivalently,
$\,y - y_1 = m(x - x_1)\,$
This gives us an extremely useful equation of a line, as summarized below:

POINT-SLOPE FORM line with slope $\,m\,$, passing through $\,(x_1,y_1)\,$
The graph of the equation $$y - y_1 = m(x - x_1)$$ is a line with slope $\,m\,$ that passes through the point $\,(x_1,y_1)\,$.
Since this equation is ideally suited to the situation where you know a point and a slope,
it is appropriately called point-slope form.
IMPORTANT THINGS TO KNOW ABOUT POINT-SLOPE FORM:
EXAMPLE:
Question:
Write the point-slope equation of the line with slope $\,5\,$ that passes through the point $\,(3,-2)\,$.
Then, write the line in $\,y = mx + b\,$ form.
Solution:
Here, $\,(x_1,y_1)\,$ is $\,(3,-2)\,$ and $\,m = 5\,$.
Substitution into $\,y - y_1 = m(x-x_1)\,$ gives: $$ y - (-2) = 5(x - 3) $$
$y$ minus known
$y$-value
equals known
slope
$($ $x$ minus known
$x$-value
$)$
$y$ $-$ $(-2)$ $=$ $5$ $($ $x$ $-$ $3$ $)$
$y$ $-$ $y_1$ $=$ $m$ $($ $x$ $-$ $x_1$ $)$


Then, put it in slope-intercept form by solving for $\,y\,$:
$\,y - (-2) = 5(x - 3)\,$(start with point-slope form)
$\,y +2 = 5x - 15\,$(simplify each side)
$\,y = 5x - 17\,$(subtract $\,2\,$ from both sides)
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Horizontal and Vertical Lines

 
 
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10
AVAILABLE MASTERED IN PROGRESS