Lesson Objectives
• Learn how to solve a two-variable system of linear equations with Cramer's rule

## How to Solve a System of Linear Equations in Two Variables Using Cramer's Rule

In this lesson, we will learn about Cramer's rule for solving a 2 x 2 linear system.

### Cramer's Rule for a 2 x 2 Linear System

$$a_{1}x + b_{1}y=c_{1}$$ $$a_{2}x + b_{2}y=c_{2}$$ First, we will set up D, this will represent the determinant of the coefficient matrix. $$D=\left| \begin{array}{cc}a_{1}&b_{1}\\ a_{2}& b_{2}\end{array}\right|$$ Next, we will set up Dx. We replace the column from D with the x coefficients with the constants. $$D_{x}=\left| \begin{array}{cc}c_{1}&b_{1}\\ c_{2}& b_{2}\end{array}\right|$$ Next, we will set up Dy. We replace the column from D with the y coefficients with the constants. $$D_{y}=\left| \begin{array}{cc}a_{1}&c_{1}\\ a_{2}& c_{2}\end{array}\right|$$ As long as D is not 0, we can find our solution for the system as: $$x=\frac{D_{x}}{D}$$ $$y=\frac{D_{y}}{D}$$ Let's look at an example.
Example #1: Solve each system. $$-2x + y=-10$$ $$-x + 4y=-5$$ Step 1) Let's find D, the determinant of the coefficient matrix: $$D=\left| \begin{array}{cc}-2&1\\ -1 & 4\end{array}\right|=-7$$ Step 2) Let's find Dx. Take D and replace the column with the x coefficients with the constants: $$D_{x}=\left| \begin{array}{cc}-10&1\\ -5 & 4\end{array}\right|=-35$$ Step 3) Let's find Dy. Take D and replace the column with the y coefficients with the constants: $$D=\left| \begin{array}{cc}-2&-10\\ -1 & -5\end{array}\right|=0$$ Step 4) Find the solution for the system: $$x=\frac{D_{x}}{D}=\frac{-35}{-7}=5$$ $$y=\frac{D_{y}}{D}=\frac{0}{-7}=0$$ Our solution for the system is given as: $$(5, 0)$$

#### Skills Check:

Example #1

Solve each system. $$-2x - 5y=11$$ $$x - 2y=8$$

A
$$(-1, 4)$$
B
$$(5, 7)$$
C
$$(2, -3)$$
D
$$(8, -1)$$
E
$$(3, -1)$$

Example #2

Solve each system. $$x + 4y=-15$$ $$-2x - 5y=15$$

A
$$(4, -7)$$
B
$$(1, 3)$$
C
$$(-2, 1)$$
D
$$(5, -5)$$
E
$$(1, -3)$$

Example #3

Solve each system. $$4x + 3y=3$$ $$4x - y=-17$$

A
$$(-1, 6)$$
B
$$(2, -4)$$
C
$$(-3, 5)$$
D
$$(-1, 1)$$
E
$$(7, 9)$$