Question

Explain the steps for solving a system of equations
using Cramer’s rule.

Short answer

For the system of equations $\left\{\begin{array}{l}{a}_{1}x+b_{1}y=k_1\\ a_{2}x+b_{2}y=k_2\end{array}\right.$, the solution $(x,y)$can be determined by, $x=\frac{D_x}{D}$,

$y=\frac{D_y}{D}$

Where, $D=\left|\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right|$

$D_x=\left|\begin{array}{cc}{k}_1& b_1\\ k_2& b_2\end{array}\right|$

$D_y=\left|\begin{array}{cc}{a}_1& k_1\\ a_2& k_2\end{array}\right|$

Step-by-step solution

Step 1. To explain

To explain the steps for solving the system of equations by using Cramer's rule.

Step 2. Consider the linear equation

Consider the system of linear equations,

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

Step 3. Find the value of D

Evaluate the determinant $D$by using the coefficients of variables.

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

$=-6-3$

$=-9$

The value of $D$is $-9$

Step 4. Find Dx

Evaluate $D_x$by replacing the coefficient of $x,-2,1$with the constants $3,12$

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

$=9-12$

$=-3$

The value of $D_x$is $-3$

Step 5. Find Dy

Evaluate $D_y$by replacing the coefficient of $y,3,3$by the constants $3,12$

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

$=-24-3$

$=-27$

The value of $D_y$is $-27$

Step 6. Find (x,y)

Find the solution $x,y$

$x=\frac{D_x}{D}$

$=\frac{-3}{-9}$

$=\frac{1}{3}$

$y=\frac{D_y}{D}$

$=\frac{-27}{-9}$

$=3$

The solution $(x,y)$is $(\frac{1}{3},3)$