Q. 30 - Precalculus Enhanced with Graphing Utilities

Question

Use Cramer’s Rule, if applicable, to solve each system.

$\left\{\begin{array}{l}2x+3y-13=0\\ 3x-2y=0\end{array}\right.$

Step-by-Step Solution

Answer

Ordered pair is $(2,3)$

Step 1. The given information is

$\left\{\begin{array}{l}2x+3y-13=0\\ 3x-2y=0\end{array}\right.$

Step 2. Finding the determinant

$$\begin{gathered} D=\left|\begin{array}{cc}2& 3\\ 3& -2\end{array}\right| \\ D=2(-2)-\left(3\right)\left(3\right) \\ D=-4-9 \\ D=-13 \\ D\ne 0 \end{gathered}$$

hence using crammer's rule 

$x=\frac{D_x}{D}, y=\frac{D_y}{D}$

Step 3. Finding D x and D y we get

$$\begin{gathered} D_x=\left|\begin{array}{cc}13& 3\\ 0& -2\end{array}\right| \\ {D}_x=13(-2)-0\left(3\right) \\ {D}_x=-26 \\ {D}_y=\left|\begin{array}{cc}2& 13\\ 3& 0\end{array}\right| \\ {D}_y=2\left(0\right)-13\left(3\right) \\ {D}_y=-39 \end{gathered}$$

Step 4. Solving for x and y we get

$$\begin{gathered} x=\frac{D_x}{D} \\ x=\frac{-26}{-13} \\ x=2 \\ y=\frac{D_y}{D} \\ y=\frac{-39}{-13} \\ y=3 \end{gathered}$$

Ordered pair is $(2,3)$