audio read-through Point-Slope Form

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Suppose a line has slope $\,m\,$ and passes through a known 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. This equation is appropriately called the point-slope form of a line.

Here's what to do:

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 $$\cssId{s22}{y - y_1 = m(x - x_1)}$$ 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:


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: $$ \cssId{s83}{y - (-2) = 5(x - 3)} $$
$y$ minus known
equals known
$($ $x$ minus known
$y$ $-$ $(-2)$ $=$ $5$ $($ $x$ $-$ $3$ $)$
$y$ $-$ $y_1$ $=$ $m$ $($ $x$ $-$ $x_1$ $)$

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

Concept Practice