Introduction

In this article, we learn how to write the equation of a line when we know:

• A point the line passes through, and
• The slope of the line.

This form is especially useful when the line does not cross the $y$‑axis at a convenient point.
We assume you already understand what slope is and how to compute it.

What is Point-Slope Form?

Point-slope form is a way to write the equation of a line using:

• A point $(x_1, y_1)$ on the line
• The slope $m$

The formula is: $$y - y_1 = m(x - x_1)$$ Key ideas:

• $(x_1, y_1)$ is any point on the line.
• $m$ tells us how steep the line is.
• This form is often easier to use than slope-intercept form.

Why Point-Slope Form Works

A line with slope $m$ changes by $m$ units vertically for every 1 unit horizontally.

So if $(x_1, y_1)$ is on the line, then any other point $(x, y)$ must satisfy: $$\frac{y - y_1}{x - x_1} = m$$ Multiplying both sides by $(x - x_1)$ gives the point-slope formula.

This is why the formula is not just convenient—it comes directly from the definition of slope.

Converting Point-Slope Form to Other Forms

Sometimes you want the equation in a different form.

To slope-intercept form ($y = mx + b$):

• Expand the right side
• Add $y_1$ to both sides
• Simplify

To standard form ($Ax + By = C$):

• Expand
• Move all terms to one side
• Clear fractions if needed

Point-slope form is flexible because it converts easily.

Examples

Example 1:

Write the equation of a line with slope $m = 4$ passing through $(2, -1)$.

Point-slope form: $$y + 1 = 4(x - 2)$$

Example 2:

Write the equation of a line with slope $m = -\frac12$ passing through $(6, 3)$. $$y - 3 = -\frac12(x - 6)$$

Example 3 (Convert to slope-intercept):

Start with: $$y - 5 = 3(x + 1)$$ Expand: $$y - 5 = 3x + 3$$ Add 5: $$y = 3x + 8$$

Exercises

1. Write the equation of a line with slope $m = 3$ passing through $(1, 4)$.

   Solution

   Point-slope form: $$y - 4 = 3(x - 1)$$

2. Write the equation of a line with slope $m = -2$ passing through $(-3, 7)$.

   Solution

   $$y - 7 = -2(x + 3)$$

3. Convert the point-slope equation $y - 2 = 5(x - 4)$ into slope-intercept form.

   Solution

   Start with: $$y - 2 = 5(x - 4)$$ Expand: $$y - 2 = 5x - 20$$ Add 2: $$y = 5x - 18$$

4. Write the equation of a line passing through $(0, -5)$ with slope $m = \frac13$.

   Solution

   $$y + 5 = \frac13(x - 0)$$ or simply: $$y + 5 = \frac13 x$$

5. A line has slope $m = -4$ and passes through $(2, 10)$. Write its equation and then convert it to standard form.

   Solution

   Point-slope: $$y - 10 = -4(x - 2)$$ Expand: $$y - 10 = -4x + 8$$ Add 10: $$y = -4x + 18$$ Standard form: $$4x + y = 18$$

6. True or false: The equation $y + 1 = 0(x - 3)$ represents a horizontal line.

   Solution

   True.
   Since $m = 0$, the line is horizontal: $$y = -1$$

7. Write the equation of a line with slope $m = \frac{5}{2}$ passing through $(-1, -1)$.

   Solution

   $$y + 1 = \frac52(x + 1)$$

8. Convert the equation $y - 6 = -3(x + 2)$ into standard form.

   Solution

   Start with: $$y - 6 = -3(x + 2)$$ Expand: $$y - 6 = -3x - 6$$ Add 6: $$y = -3x$$ Standard form: $$3x + y = 0$$