Question

In the following exercises, solve each system of equations using Cramer’s rule

$\left\{\begin{array}{l}x-3y=-9\\ 2x+5y=4\end{array}\right.$

Short answer

The ordered pair$\left(-3,2\right)$is the solution for the given system of equations.

Step-by-step solution

Step 1. Given information

The given system of equations is:

$\left\{\begin{array}{l}x-3y=-9\\ 2x+5y=4\end{array}\right.$

Step 2. Evaluating the determinant D

In determinant $D$all the coefficients are taken.

So,

$D=\left|\begin{array}{cc}1& -3\\ 2& 5\end{array}\right|$

$$\begin{gathered} D=1\times 5-(-3\times 2) \\ D=5+6 \\ D=11 \end{gathered}$$

Step 3. Evaluating the determinant Dx

In determinant $D_x$, we take the constants in place of coefficients of $x$

So.

$$\begin{gathered} D_x=\left|\begin{array}{cc}-9& -3\\ 4& 5\end{array}\right| \\ {D}_x=-9\times 5-(-3\times 4) \\ {D}_x=-45+12 \\ {D}_x=-33 \end{gathered}$$

Step 4. Evaluating the determinant Dy

In the determinant $D_y$, we take the constant in place of coefficients of $y$

So,

$$\begin{gathered} D_y=\left|\begin{array}{cc}1& -9\\ 2& 4\end{array}\right| \\ {D}_y=1\times 4-(-9\times 2) \\ {D}_y=4+18 \\ {D}_y=22 \end{gathered}$$

Step 5. Finding x and y

For $x$

$$\begin{gathered} x=\frac{D_x}{D} \\ x=\frac{-33}{11} \\ x=-3 \end{gathered}$$

For $y$

$$\begin{gathered} y=\frac{D_y}{D} \\ y=\frac{22}{11} \\ y=2 \end{gathered}$$

Step 6. Writing solution in ordered pair

The solution of the system in ordered pair is$\left(-3,2\right)$

Step 7. Check the solution for x − 3y = −9

Substituting $$\begin{gathered} x=-3 \\ y=2 \end{gathered}$$in the equation, we get:

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

This is true

Step 8. Check the solution for 2x + 5y = 4

Substituting $$\begin{gathered} x=-3 \\ y=2 \end{gathered}$$in the equation, we get:

$$\begin{gathered} 2x+5y=4 \\ 2\left(-3\right)+5\left(2\right)=4 \\ -6+10=4 \\ 4=4 \end{gathered}$$

This is also true.

Thus, the ordered pair $(-3,2)$is the solution.