# Derive Point-Slope Formula

#### harpazo

##### Pure Mathematics
Banned
How do we derive the point-slope formula in the picture?

#### MarkFL

##### La Villa Strangiato
Staff member
Moderator
Math Helper
Slope is defined as:

$$\displaystyle m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$$

If we let the second point be the arbitrary point $$(x,y)$$, we then have:

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

This gives us the locus of all points that have a slope of $$m$$ with respect to $$\left(x_1,y_1\right)$$, and it can be arranged as:

$$\displaystyle y-y_1=m\left(x-x_1\right)$$

harpazo

#### harpazo

##### Pure Mathematics
Banned
Slope is defined as:

$$\displaystyle m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$$

If we let the second point be the arbitrary point $$(x,y)$$, we then have:

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

This gives us the locus of all points that have a slope of $$m$$ with respect to $$\left(x_1,y_1\right)$$, and it can be arranged as:

$$\displaystyle y-y_1=m\left(x-x_1\right)$$
How often does this formula pop up in calculus 1?

Staff member