Question

Solve using Cramer’s rule.

$\left\{\begin{array}{l}3x+y=-3\\ 2x+3y=6\end{array}\right.$

Short answer

Solution for system of linear equation is$\left(-\frac{15}{7},\frac{24}{7}\right)$

Step-by-step solution

Step 1. Given information

Given system of linear equation,

$\begin{array}{l}3x+y=-3\\ 2x+3y=6\end{array}$

Step 2. Determining the solution

Variable coefficients are needed to get the determinant.

i.e.

$D=\left|\begin{array}{cc}3& 1\\ 2& 3\end{array}\right|$

Let's evaluate the determinant.

$$\begin{gathered} D=\left|\begin{array}{cc}3& 1\\ 2& 3\end{array}\right| \\ =\left(3\right)\left(3\right)-\left(1\right)\left(2\right) \\ =9-2 \\ =7 \end{gathered}$$

Step 3. Finding the solution.

Let's evaluate $D_x$by using the constant in place of x coefficients.

$$\begin{gathered} D_x=\left|\begin{array}{cc}-3& 1\\ 6& 3\end{array}\right| \\ =(-3)\left(3\right)-\left(1\right)\left(6\right) \\ =-9-6 \\ =-15 \end{gathered}$$

Let's evaluate $D_y$by using the constant in place of y coefficients.

$$\begin{gathered} D_y=\left|\begin{array}{cc}3& -3\\ 2& 6\end{array}\right| \\ =\left(3\right)\left(6\right)-(-3)\left(2\right) \\ =18+6 \\ =24 \end{gathered}$$

Step 4. Find x and y.

Put the values of $D,D_{x}and{D}_y$

To find x:

$$\begin{gathered} x=\frac{D_x}{D} \\ =-\frac{15}{7} \end{gathered}$$

To find y:

$$\begin{gathered} y=\frac{D_y}{D} \\ =\frac{24}{7} \end{gathered}$$