3.3 Find the Equation of a Line

Linear equations describe how two variables relate to each other, and writing the equation of a line is one of the most fundamental skills in algebra. This section covers how to find a line's equation when you're given different starting information: a slope and y-intercept, a slope and a point, two points, or a relationship to another line (parallel or perpendicular).

Equations of Lines

Line equation from slope and y-intercept

This is the most straightforward case. If you already know the slope and y-intercept, you just plug them into slope-intercept form:

$y = mx + b$

• $m$ is the slope, which tells you the steepness and direction of the line. A slope of $m = 2$ means the line rises 2 units for every 1 unit it moves to the right.
• $b$ is the y-intercept, the y-coordinate where the line crosses the y-axis. If $b = 3$, the line passes through the point $(0, 3)$.

To write the equation, substitute your values for $m$ and $b$ directly.

Example: Given slope $m = -\frac{1}{2}$ and y-intercept $b = 4$:

$y = -\frac{1}{2}x + 4$

That's it. No simplifying needed.

Line equation from slope and a point

When you know the slope and one point on the line (but not the y-intercept), use point-slope form:

$y - y_1 = m(x - x_1)$

Here's the process:

1. Identify the slope $m$ and the point $(x_1, y_1)$.
2. Substitute those values into point-slope form.
3. Distribute the slope on the right side.
4. Solve for $y$ to convert to slope-intercept form.

Example: Given slope $m = 3$ and point $(2, 5)$:

• Start with point-slope form: $y - 5 = 3(x - 2)$
• Distribute: $y - 5 = 3x - 6$
• Add 5 to both sides: $y = 3x - 1$

The final equation is $y = 3x - 1$, where $m = 3$ and $b = -1$.

Line equation from two points

If you're given two points but no slope, you need to calculate the slope first, then use point-slope form.

1. Find the slope using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

2. Pick either point (both will give the same final equation).

3. Plug the slope and your chosen point into point-slope form.

4. Simplify to slope-intercept form.

Example: Given points $(1, 2)$ and $(4, 8)$:

• Calculate slope: $m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$
• Use point $(1, 2)$: $y - 2 = 2(x - 1)$
• Distribute: $y - 2 = 2x - 2$
• Add 2 to both sides: $y = 2x$

The equation is $y = 2x$, with slope $m = 2$ and y-intercept $b = 0$ (the line passes through the origin).

A common mistake here is subtracting the coordinates in the wrong order. Make sure you're consistent: if you start with $y_2$, you must start with $x_2$ in the denominator too.

Equation of a parallel line

Two lines are parallel if and only if they have the same slope (but different y-intercepts). So to find a line parallel to a given line through a specific point:

1. Identify the slope of the given line.
2. Use that same slope with the new point in point-slope form.
3. Simplify to slope-intercept form.

Example: Find the line parallel to $y = 2x + 3$ that passes through $(1, 6)$.

• The given line has slope $m = 2$. The parallel line also has slope $m = 2$.
• Point-slope form: $y - 6 = 2(x - 1)$
• Distribute: $y - 6 = 2x - 2$
• Add 6: $y = 2x + 4$

Notice both lines have slope 2, but different y-intercepts (3 vs. 4), confirming they're parallel.

Equation of a perpendicular line

Two lines are perpendicular if their slopes are negative reciprocals of each other. That means you flip the fraction and change the sign. If one slope is $m_1$, the perpendicular slope is:

$m_2 = -\frac{1}{m_1}$

Here are some quick examples of negative reciprocals:

• Slope of $3$ → perpendicular slope of $-\frac{1}{3}$
• Slope of $-\frac{2}{5}$ → perpendicular slope of $\frac{5}{2}$
• Slope of $1$ → perpendicular slope of $-1$

To find the equation:

1. Find the slope of the given line.
2. Take the negative reciprocal to get the perpendicular slope.
3. Use the perpendicular slope and the given point in point-slope form.
4. Simplify.

Example: Find the line perpendicular to $y = 3x - 2$ that passes through $(2, 1)$.

• The given slope is $m_1 = 3$, so the perpendicular slope is $m_2 = -\frac{1}{3}$.
• Point-slope form: $y - 1 = -\frac{1}{3}(x - 2)$
• Distribute: $y - 1 = -\frac{1}{3}x + \frac{2}{3}$
• Add 1 (which is $\frac{3}{3}$) to both sides: $y = -\frac{1}{3}x + \frac{5}{3}$

Watch the fractions carefully in that last step. Adding $\frac{2}{3} + \frac{3}{3} = \frac{5}{3}$ is where many students make arithmetic errors.

Additional Concepts

• The x-intercept is the point where a line crosses the x-axis. You can find it by setting $y = 0$ and solving for $x$. For example, in $y = 2x + 4$, setting $y = 0$ gives $x = -2$, so the x-intercept is $(-2, 0)$.
• If a line's equation is not already in slope-intercept form, you may need to rearrange it before identifying the slope. For instance, if you're told a line has equation $4x + 2y = 6$, solve for $y$ first: $y = -2x + 3$. Now you can see the slope is $-2$.
• Linear equations model many real-world relationships, such as cost vs. quantity, distance vs. time, or temperature conversions. The slope represents the rate of change, and the y-intercept represents the starting value.