Point Slope Form

Point-slope form is the line formula

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

Use it when you know one point on a non-vertical line and the slope. In this formula, $(x_1, y_1)$ is the known point and $m$ is the slope. It is often the fastest way to write the equation before converting to slope-intercept form.

What Point-Slope Form Means

Slope compares vertical change to horizontal change. If a line has slope $m$, then

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

as long as $x \ne x_1$. Multiplying both sides by $x - x_1$ gives

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

So point-slope form is just the slope definition rewritten so the known point stays visible.

Why The Formula Is Useful

Think of $(x_1, y_1)$ as an anchor point. The expression $x - x_1$ tells you how far you moved horizontally from that point. Multiplying by $m$ tells you the matching vertical change, so $y - y_1$ has to equal $m(x - x_1)$.

That is why this form feels direct: start from one known point, then build the line using its slope.

Worked Example: Write A Line From A Point And A Slope

Find the equation of the line with slope $-4$ that passes through $(2, 3)$.

Start with the formula:

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

Substitute $m = -4$, $x_1 = 2$, and $y_1 = 3$:

$$y - 3 = -4(x - 2)$$

That is already a correct final answer in point-slope form.

If you want slope-intercept form, expand:

$$y - 3 = -4x + 8$$ $$y = -4x + 11$$

Both equations describe the same line. Point-slope form and slope-intercept form are different ways to write the same relationship.

A quick check keeps mistakes from slipping through. Plug in the given point:

$$y - 3 = -4(2 - 2) = 0$$

So $y = 3$, which matches the original point $(2, 3)$.

Common Point-Slope Form Mistakes

1. Reversing the point values. If the point is $(2, 3)$, write $y - 3$ and $x - 2$, not $y - 2$ and $x - 3$.
2. Losing the minus sign with negative coordinates. If the point is $(-1, 5)$, then $x - (-1)$ becomes $x + 1$.
3. Thinking the equation must be simplified. $y - 3 = -4(x - 2)$ is already a valid line equation.
4. Using point-slope form for a vertical line. A vertical line has undefined slope, so it is written as $x = c$ instead.

When To Use Point-Slope Form

Use point-slope form when both of these are known:

1. One point on a non-vertical line
2. The slope of that line

It shows up often in algebra and coordinate geometry problems because many questions give exactly that information. It is also useful after you compute a slope from two points and still need the line equation.

A Fast Check Before You Move On

Look back at the point the problem gave you. If you cannot clearly see that point inside $y - y_1 = m(x - x_1)$, or if plugging it in does not make both sides equal, the substitution is probably off.

When This Form Shines

Point-slope form, $y - y_1 = m(x - x_1)$, is the natural choice the moment you know one point and the slope, which is exactly the information most line problems hand you. Write the equation in this form first, and convert to slope-intercept form only if a problem specifically asks for it. The two forms describe the same line; they just emphasize different starting information.