Lesson 5: Point-Slope Formula

New Concept

The point-slope formula, $y - y_1 = m(x - x_1)$, is a key tool for defining a line. It allows you to find a line's equation and create its graph using its slope ($m$) and any single point on it.

What’s next

Now, let's put this formula into action. You'll work through guided examples and practice problems to find and graph linear equations.

Property

To find an equation for the line of slope $m$ passing through the point $(x_1, y_1)$, we use the point-slope formula

or

Examples

• To find the equation of a line with slope $m=4$ passing through $(2, 7)$, we use the formula: $y - 7 = 4(x - 2)$, which simplifies to $y = 4x - 1$.
• Find an equation for the line passing through $(6, 2)$ that is perpendicular to $y = 3x + 5$. The perpendicular slope is $m = \frac{-1}{3}$. The equation is $y - 2 = \frac{-1}{3}(x - 6)$, which simplifies to $y = \frac{-1}{3}x + 4$.
• A line is parallel to $y = -5x + 1$ and passes through $(-1, 3)$. The slope is $m = -5$. The equation is $y - 3 = -5(x - (-1))$, which simplifies to $y = -5x - 2$.

Explanation

This formula is your go-to tool for finding a line's equation when you know its slope and any single point it passes through. It directly uses the given point, which is often easier than finding the y-intercept first.

Property

They are really the same formula, but they are used for different purposes:

• We use the slope formula to calculate the slope of a line when we know two points on the line. That is, we know $(x_1, y_1)$ and $(x_2, y_2)$, and we are looking for $m$. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• We use the point-slope formula to find the equation of a line. That is, we know $(x_1, y_1)$ and $m$, and we are looking for an equation like $y = mx + b$. The point-slope formula is $m = \frac{y - y_1}{x - x_1}$.

Examples

• To find the slope of a line passing through $(3, 6)$ and $(5, 12)$, use the slope formula: $m = \frac{12 - 6}{5 - 3} = \frac{6}{2} = 3$.
• A line has a slope of $5$ and passes through $(-1, 4)$. To find its equation, use the point-slope formula: $y - 4 = 5(x - (-1))$, which simplifies to $y = 5x + 9$.
• To find the equation of a line through $(2, 1)$ and $(4, 7)$, first find the slope $m = \frac{7-1}{4-2} = 3$. Then use the point-slope formula with $(2, 1)$: $y - 1 = 3(x - 2)$.

Explanation

The slope formula finds the value of the slope, $m$, using two known points. The point-slope formula uses that slope and one point to build the entire equation of the line. Think of it as finding the tool, then using the tool.

Property

To Fit a Line Through Two Points.

1. Compute the slope between the two points.
2. Substitute the slope and either point into the point-slope formula.

Examples

• Find the equation for the line that passes through $(1, 4)$ and $(3, 10)$. First, find the slope: $m = \frac{10 - 4}{3 - 1} = 3$. Now use the point-slope formula: $y - 4 = 3(x - 1)$.
• Find the equation for the line through $(-2, 5)$ and $(1, -1)$. The slope is $m = \frac{-1 - 5}{1 - (-2)} = \frac{-6}{3} = -2$. Using the point-slope formula gives $y - 5 = -2(x + 2)$.
• Find the equation for the line through $(5, -3)$ and $(-1, 0)$. The slope is $m = \frac{0 - (-3)}{-1 - 5} = \frac{3}{-6} = \frac{-1}{2}$. Using the point-slope formula gives $y - 0 = \frac{-1}{2}(x + 1)$.

Explanation

Only one unique straight line can pass through any two given points. This method provides a reliable two-step process to find that line's equation: first, calculate the slope, then plug it into the point-slope formula with either point.

Property

Variables that increase or decrease at a constant rate can be described by linear equations. To model this, treat two related data pairs as points $(x_1, y_1)$ and $(x_2, y_2)$. First, compute the slope (rate of change), then substitute the slope and either point into the point-slope formula to find the governing equation.

Examples

• A taxi ride costs 10 dollars for 2 miles and 16 dollars for 4 miles. Let cost be $C$ and distance be $d$. The points are $(2, 10)$ and $(4, 16)$. The slope (cost per mile) is $m = \frac{16-10}{4-2} = 3$. The equation is $C - 10 = 3(d - 2)$.
• A tree was 8 feet tall in 2015 and 14 feet tall in 2018. Let height be $H$ and the year be $t$ (with $t=0$ in 2015). The points are $(0, 8)$ and $(3, 14)$. The slope is $m = \frac{14-8}{3-0} = 2$ feet per year. The equation is $H = 2t + 8$.
• A phone plan costs 40 dollars for 5 GB of data and 50 dollars for 10 GB. Let cost be $C$ and data be $D$. The points are $(5, 40)$ and $(10, 50)$. The slope is $m = \frac{50-40}{10-5} = 2$ dollars per GB. The equation is $C - 40 = 2(D - 5)$.

Explanation

Real-world scenarios with a steady rate of change can be modeled using a linear equation. This allows you to make predictions by finding the line's equation from just two data points, like cost over time or distance versus speed.