Forms of the equation of a line

What you will be able to do

• write the equation of a horizontal or a vertical line;
• write the equation of a line in point-slope, two-point, slope-intercept and intercept form;
• find the intercepts of a line and the slope from its intercepts;
• find the point that divides a segment in a given ratio, and use it to fix a line.

Horizontal and vertical lines

Every point of a horizontal line has the same $y$-coordinate. A horizontal line through $(0, b)$ is

A vertical line has the same $x$-coordinate everywhere. The vertical line through $(a, 0)$ is

The $x$-axis is $y = 0$ and the $y$-axis is $x = 0$.

Point-slope form

Let a line pass through $(x_1, y_1)$ with slope $m$. A point $P(x, y)$ is on the line when the slope from $(x_1, y_1)$ to $P$ is also $m$:

Two-point form

Suppose the line passes through $(x_1, y_1)$ and $(x_2, y_2)$. Its slope is $\dfrac{y_2 - y_1}{x_2 - x_1}$, and the point-slope form gives

Slope-intercept form

Suppose a line has slope $m$ and cuts the $y$-axis at $(0, c)$. The number $c$ is the $y$-intercept. The point-slope form at $(0, c)$ gives

A vertical line cannot be written this way, because it has no slope.

Intercept form

Suppose a line cuts the $x$-axis at $(a, 0)$ and the $y$-axis at $(0, b)$. The numbers $a$ and $b$ are its intercepts. The slope is $\dfrac{b - 0}{0 - a} = -\dfrac{b}{a}$, so $y - b = -\dfrac{b}{a}(x - 0)$. Divide by $b$ and rearrange:

This form needs $a \neq 0$ and $b \neq 0$. It cannot describe a line through the origin, or a line parallel to an axis.

A point that divides a segment

Sometimes a line has to pass through a point defined by a ratio. Let $P$ lie on the segment from $A(x_1, y_1)$ to $B(x_2, y_2)$, with $AP : PB = m : n$. Then

The larger part of the ratio multiplies the far point. When $m = n$, the formula gives the midpoint.

Worked examples

The line through two points. Find the equation of the line through $A(3, 1)$ and $B(7, 9)$, and its intercepts.

1. The slope. $m = \dfrac{9 - 1}{7 - 3} = 2$.
2. Point-slope form. $y - 1 = 2(x - 3)$, so $y = 2x - 5$.
3. The intercepts. At $x = 0$, $y = -5$. At $y = 0$, $x = 2.5$.

A point from a ratio. The point $P$ divides the segment from $A(-1, 2)$ to $B(5, 8)$ in the ratio $1 : 2$. Find the line through $P$ with slope $-3$.

1. The $x$-coordinate. $\dfrac{1 \times 5 + 2 \times (-1)}{3} = 1$.
2. The $y$-coordinate. $\dfrac{1 \times 8 + 2 \times 2}{3} = 4$, so $P = (1, 4)$.
3. The line. $y - 4 = -3(x - 1)$, so $y = -3x + 7$.

Intercepts in a ratio. A line passes through $(6, 1)$, and its $x$-intercept is $3$ times its $y$-intercept. Find the intercepts.

1. The form. Let the $y$-intercept be $b$. Then $\dfrac{x}{3b} + \dfrac{y}{b} = 1$.
2. Use the point. $\dfrac{6}{3b} + \dfrac{1}{b} = 1$, so $\dfrac{3}{b} = 1$ and $b = 3$.
3. The answer. The $y$-intercept is $3$ and the $x$-intercept is $9$.

Practise with new numbers

Try it in Python

The program finds the equation of a line from two points, then its intercepts, and checks that both points satisfy it.

Common mistakes

• Writing $x + x_1$ in point-slope form. The form is $y - y_1 = m(x - x_1)$, with minus signs in both brackets.
• Using intercept form for a line through the origin. Both intercepts are $0$, and the form needs a division by $0$.
• Writing a vertical line as $y = mx + c$. A vertical line has the equation $x = a$.
• Mixing up the intercept with the slope. In $y = mx + c$, the slope is $m$ and the $y$-intercept is $c$.
• Swapping the ratio. In the section formula, the part $m$ multiplies the far point $B$.

Remember these

• Horizontal: $y = b$. Vertical: $x = a$.
• Point-slope: $y - y_1 = m(x - x_1)$. Slope-intercept: $y = mx + c$.
• Two-point: $y - y_1 = \dfrac{y_2 - y_1}{x_2 - x_1}(x - x_1)$.
• Intercept form: $\dfrac{x}{a} + \dfrac{y}{b} = 1$, with slope $-\dfrac{b}{a}$.
• The point dividing $AB$ in the ratio $m : n$ is $\left(\dfrac{m x_2 + n x_1}{m + n}, \dfrac{m y_2 + n y_1}{m + n}\right)$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Write the equation of the line through $(2, -4)$ with slope $5$.

2. Find the equation of the line through $(1, 2)$ and $(3, 8)$.

3. Write the line with intercepts $4$ and $-3$ in intercept form, and find its slope.

4. Write the equation of the vertical line, and of the horizontal line, through $(5, -2)$.

5. Where does the line of slope $2$ through $(3, 1)$ cut the $x$-axis?