4.6 Find the Equation of a Line

Equations of Lines

Line equation from slope and y-intercept

The most direct way to write a line's equation is when you already know the slope and the y-intercept. You'll use slope-intercept form:

$y = mx + b$

• $m$ is the slope (steepness and direction of the line). Positive slopes rise left to right; negative slopes fall.
• $b$ is the y-intercept, the point where the line crosses the y-axis at $(0, b)$.

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

Example: slope = 2, y-intercept = -3

$y = 2x + (-3)$, which simplifies to $y = 2x - 3$

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:

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

• $m$ is the slope.
• $(x_1, y_1)$ is the known point on the line.

Steps:

1. Substitute the slope for $m$ and the point's coordinates for $x_1$ and $y_1$.
2. If the problem asks for slope-intercept form, distribute the slope and solve for $y$.

Example: slope = $-\frac{1}{2}$, point $(4, 6)$

1. Substitute: $y - 6 = -\frac{1}{2}(x - 4)$

2. Distribute: $y - 6 = -\frac{1}{2}x + 2$

3. Add 6 to both sides: $y = -\frac{1}{2}x + 8$

Both the point-slope version and the simplified slope-intercept version are correct equations for the same line. Your instructor may specify which form to give.

Line equation from two points

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

Steps:

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

2. Pick either point and substitute it along with the slope into point-slope form.

3. Simplify if needed.

Example: points $(1, 3)$ and $(5, 11)$

1. Calculate slope: $m = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2$

2. Use the point $(1, 3)$: $y - 3 = 2(x - 1)$

3. Simplify: $y - 3 = 2x - 2$, so $y = 2x + 1$

You can plug in either point. Using $(5, 11)$ instead gives $y - 11 = 2(x - 5)$, which simplifies to the same equation: $y = 2x + 1$.

Equation of a parallel line

Parallel lines have the same slope but different y-intercepts. They never intersect because they rise and run at the same rate, just shifted vertically.

To write the equation of 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.

Example: Find the line parallel to $y = 3x - 2$ that passes through $(1, 5)$.

1. The given slope is 3. The parallel line also has slope 3.

2. Substitute into point-slope form: $y - 5 = 3(x - 1)$

3. Simplify: $y = 3x + 2$

Notice the two lines share the slope of 3 but have different y-intercepts (-2 vs. 2).

Equation of a perpendicular line

Perpendicular lines meet at a 90° angle. Their slopes are negative reciprocals of each other, meaning you flip the fraction and change the sign. The product of perpendicular slopes always equals -1.

To find the negative reciprocal: if the original slope is $m$, the perpendicular slope is $-\frac{1}{m}$.

Steps:

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

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

1. The given slope is $-2$. The negative reciprocal is $\frac{1}{2}$.

2. Substitute: $y - (-1) = \frac{1}{2}(x - 2)$

3. Simplify: $y + 1 = \frac{1}{2}x - 1$, so $y = \frac{1}{2}x - 2$

Quick check: $(-2) \times \frac{1}{2} = -1$. The slopes multiply to -1, confirming they're perpendicular.

Linear Functions and Graphs

A linear function produces a straight line when graphed. Its defining feature is a constant rate of change, which is the slope. Every point on the line satisfies the equation, and every solution to the equation is a point on the line.

You can write linear equations in different forms depending on what information you have:

• Slope-intercept form $y = mx + b$: best when you know the slope and y-intercept.
• Point-slope form $y - y_1 = m(x - x_1)$: best when you know the slope and any point.

These forms describe the same line, just written differently. Converting between them is a matter of algebraic simplification.