Point-Slope Form

By the end of this lesson, you will be able to write a line's equation from any point and a slope, convert between point-slope and slope-intercept form, and say why this form exists when you already had another one.

The Form Built for What You Are Usually Given

$$y - y_1 = m(x - x_1)$$

A word problem almost never hands you the $y$-intercept. It hands you a slope and some point that is not on the axis at all: "the plumber charges $$45$ an hour and a 3-hour job cost $$210$."

With slope-intercept you would have to substitute, solve for $b$, and then write the equation. Point-slope skips that. Drop the numbers straight in:

$$y - 210 = 45(x - 3)$$

Done, in one line, with no algebra performed. That is the entire reason the form exists.

Where It Comes From

It is the slope formula, rearranged, and seeing that makes it impossible to forget.

Take any point $(x, y)$ on the line and a known point $(x_1, y_1)$. The slope between them is $m$:

$$\frac{y - y_1}{x - x_1} = m$$

Multiply both sides by $(x - x_1)$:

$$y - y_1 = m(x - x_1)$$

So the form is not a new fact. It is the definition of slope with the fraction cleared, which is why the subtractions are the way round they are.

Watch the Signs

The form subtracts both coordinates, so a negative coordinate becomes a double negative and turns into a plus.

Point	Substituted	Tidied
$(3, 5)$	$y - 5 = m(x - 3)$	unchanged
$(-2, 5)$	$y - 5 = m(x - (-2))$	$y - 5 = m(x + 2)$
$(3, -4)$	$y - (-4) = m(x - 3)$	$y + 4 = m(x - 3)$
$(-2, -4)$	$y + 4 = m(x + 2)$	both flip

Substitute with the brackets in place first, then tidy. Trying to do the sign in your head is where this form goes wrong, and it is the only place it goes wrong.

Example 1: Straight from a Point and a Slope

Here is the whole method on one line, worked all the way through. Write the line through $(4, 7)$ with slope $3$.

• Substitute into y - y1 = m(x - x1), keeping the brackets y - 7 = 3(x - 4)
• Multiply out the bracket y - 7 = 3x - 12
• Add 7 to both sides to get slope-intercept form y = 3x - 5

Worked Example 1: A Negative Coordinate

Write the line through $(-2, 5)$ with slope $-4$.

From Two Points

Given two points and no slope, work the slope out first and then use either point. Either one gives the same line, which is worth proving to yourself once.

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