A Point and a Slope

A line is fixed by one point on it and its slope. Point-slope form writes the equation of the line through the point (x1, y1) with slope m:

$$y-y\cdot 1 = m\left(x-x\cdot 1\right)$$

For the line through (1, 2) with slope 3, substitute the point and the slope: $y-2 = 3\left(x-1\right)$. To change it into slope-intercept form, expand the bracket to get $y-2 = 3x-3$. Then add 2 to both sides to get $y = 3x-1$.

Point-slope form comes from the slope formula. Take any point (x, y) on the line. The rise from (x1, y1) to (x, y) is $y-y\cdot 1$ and the run is $x-x\cdot 1$, so $\frac{y-y\cdot 1}{x-x\cdot 1} = m$. Multiply both sides by the run $x-x\cdot 1$ to get $y-y\cdot 1 = m\left(x-x\cdot 1\right)$. To use the form, substitute the known point and the slope, then expand the bracket.