Lesson Objectives
• Learn how to find the equation of a line using determinants

## How to Find the Equation of a Line Using Determinants

In this lesson, we will learn how to find the equation of a line using determinants. Over the course of the last two lessons, we have been working with a formula that finds the area of a triangle. If the result of this formula is zero, our area is zero, therefore, our three points are on the same line or are collinear. We can use this fact to write the equation of a line passing through two points: $$(x_1, y_1), (x_2, y_2)$$ $$\left| \begin{array}{ccc}x & y&1\\ x_{1}& y_{1}& 1\\ x_{2}& y_{2}& 1\end{array}\right|=0$$ We can apply the determinant formula using Laplace expansion to get our equation.

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$$x \left| \begin{array}{cc}y_1 & 1\\ y_2& 1\end{array}\right|-y\left| \begin{array}{cc}x_1 & 1\\ x_2& 1\end{array}\right|+\left| \begin{array}{cc}x_1 & y_1\\ x_2& y_2\end{array}\right|=0$$
Let's look at an example.
Example #1: Find the slope-intercept form of the line. $$(-2, 4), (0, 3)$$ Plug into the formula: $$\text{Point 1}: (-2, 4)$$ $$\text{Point 2}: (0, 3)$$ $$\left| \begin{array}{ccc}x & y&1\\ -2& 4& 1\\ 0& 3& 1\end{array}\right|=0$$ We will use the formula determinant formula:

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$$x \left| \begin{array}{cc}4 & 1\\ 3& 1\end{array}\right|-y\left| \begin{array}{cc}-2 & 1\\ 0& 1\end{array}\right|+\left| \begin{array}{cc}-2 & 4\\ 0& 3\end{array}\right|=0$$
$$1x + 2y - 6=0$$ Solve for y: $$2y=-x + 6$$ $$y=-\frac{1}{2}x + 3$$

#### Skills Check:

Example #1

Find the slope-intercept form of the line. $$(0, -1), (-5, 0)$$

A
$$y=\frac{5}{2}x + 3$$
B
$$y=-\frac{1}{5}x - 1$$
C
$$y=\frac{2}{3}x + 5$$
D
$$y=\frac{7}{5}x + 10$$
E
$$y=\frac{12}{7}x + 4$$

Example #2

Find the slope-intercept form of the line. $$(-3, 2), (0, -2)$$

A
$$y=-\frac{4}{3}x - 2$$
B
$$y=-\frac{3}{4}x + 4$$
C
$$y=\frac{5}{7}x + 1$$
D
$$y=-\frac{1}{2}x + 7$$
E
$$y=-\frac{3}{4}x + 2$$

Example #3

Find the slope-intercept form of the line. $$(3, -3), (-1, -5)$$

A
$$y=4x + 3$$
B
$$y=\frac{1}{3}x + 1$$
C
$$y=\frac{5}{7}x + 7$$
D
$$y=\frac{1}{2}x - \frac{9}{2}$$
E
$$y=\frac{3}{5}x + 2$$