If you know one point on a line and its slope, you know the whole line. Point-slope form writes that fact down directly, with no need to find the $y$-intercept first. It builds on what you know about graphs as solutions.
Por qué funciona
Let $(x_1, y_1)$ be the known point and $(x, y)$ any other point on the line. The slope between them must be $m$: $$\frac{y - y_1}{x - x_1} = m$$ Multiply both sides by $(x - x_1)$ and you get point-slope form, $y - y_1 = m(x - x_1)$. Every point on the line satisfies it, and so does the known point itself (both sides are $0$).
Idea clave
The line through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$. Solve for $y$ if you want slope-intercept form.
Worked examples
Ejemplo resuelto: Write a line
Write an equation of the line through $(2, -3)$ with slope $4$.
- Substitute $m = 4$, $x_1 = 2$, $y_1 = -3$: $y - (-3) = 4(x - 2)$.
- Simplify the signs: $y + 3 = 4(x - 2)$. This is a complete answer.
- For slope-intercept form, distribute and subtract $3$: $y = 4x - 11$.
Respuesta: $y + 3 = 4(x - 2)$, or $y = 4x - 11$.
Ejemplo resuelto: Read the point and slope
Which point and slope does $y - 2 = -\frac{1}{2}(x + 5)$ describe?
- Match it to $y - y_1 = m(x - x_1)$. $y - 2$ gives $y_1 = 2$.
- $x + 5 = x - (-5)$, so $x_1 = -5$.
- The slope is the factor in front: $m = -\frac{1}{2}$.
Respuesta: The point $(-5, 2)$ and slope $-\frac{1}{2}$.
Error común
Reading the point from $y - 2 = 3(x + 1)$ as $(1, 2)$. The form subtracts the coordinates, so $x + 1$ means $x_1 = -1$, and the point is $(-1, 2)$. Check: put $x = -1$ and $y = 2$ in; both sides should be $0$.