Question

Solve system of equations using Cramer’s Rule.

$\left\{\begin{array}{l}-2x+3y=3\\ x+3y=12\end{array}\right.$

Short answer

The solution for the system of linear equation is$(3,3)$

Step-by-step solution

Step 1. Given information

The system of linear equations is$\left\{\begin{array}{l}-2x+3y=3\\ x+3y=12\end{array}\right.$

Step 2. Find the determinant D and evaluate it

We will take the coefficients of the variables to form the determinant $D$

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

Evaluate it,

$$\begin{gathered} D=(-2)\left(3\right)-\left(3\right)\left(1\right) \\ D=-6-3 \\ D=-9 \end{gathered}$$

Step 3. Find the determinant Dx and evaluate it

We will use the constants in place of the $x$coefficients to find the determinant $D_x$

$D_x=\left|\begin{array}{cc}3& 3\\ 12& 3\end{array}\right|$

Evaluate it

$$\begin{gathered} D_x=\left(3\right)\left(3\right)-\left(3\right)\left(12\right) \\ {D}_x=9-36 \\ {D}_x=-27 \end{gathered}$$

Step 4. Find the determinant Dy and evaluate it

We will use the constants in place of the $y$coefficients to find the determinant$D_y$

$D_y=\left|\begin{array}{cc}-2& 3\\ 1& 12\end{array}\right|$

Evaluate it

$$\begin{gathered} D_y=(-2)\left(12\right)-\left(3\right)\left(1\right) \\ {D}_y=-24-3 \\ {D}_y=-27 \end{gathered}$$

Step 5. Find x and y

To find $x$,

$$\begin{gathered} x=\frac{D_x}{D} \\ x=\frac{-27}{-9} \\ x=3 \end{gathered}$$

To find $y$,

$$\begin{gathered} y=\frac{D_y}{D} \\ y=\frac{-27}{-9} \\ y=3 \end{gathered}$$

The solution for the system of linear equation is$(3,3)$

Step 6. Check by substituting values 

Substitute $x=3,y=3$in both the equations,

$$\begin{gathered} x+3y=12 \\ 3+3\left(3\right)=12 \\ 3+9=12 \\ 12=12 \end{gathered}$$

This is true.

$$\begin{gathered} -2x+3y=3 \\ -2\left(3\right)+3\left(3\right)=3 \\ -6+9=3 \\ 3=3 \end{gathered}$$

This is also true.

Therefore, the solution is $(3,3)$