Q 249 - Intermediate Algebra

Question

Solve system of equations using Cramer’s Rule. 

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

Step-by-Step Solution

Answer

The solution for the system of linear equation is $(7,6)$

Step 1. Given information

The system of linear equations is $\left\{\begin{array}{l}x-2y=-5\\ 2x-3y=-4\end{array}\right.$

Step 2. Find the determinant D and evaluate it

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

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

Evaluate it,

$$\begin{gathered} D=\left(1\right)(-3)-(-2)\left(2\right) \\ D=-3+4 \\ D=1 \end{gathered}$$

Step 3. Find the determinant D x and evaluate it

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

$D_x=\left|\begin{array}{cc}-5& -2\\ -4& -3\end{array}\right|$

Evaluate it

$$\begin{gathered} D_x=(-5)(-3)-(-2)(-4) \\ {D}_x=15-8 \\ {D}_x=7 \end{gathered}$$

Step 4. Find the determinant D y and evaluate it

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

$D_y=\left|\begin{array}{cc}1& -5\\ 2& -4\end{array}\right|$

Evaluate it

$$\begin{gathered} D_y=\left(1\right)(-4)-(-5)\left(2\right) \\ {D}_y=-4+10 \\ {D}_y=6 \end{gathered}$$

Step 5. Find x and y

To find $x$,

$$\begin{gathered} x=\frac{D_x}{D} \\ x=\frac{7}{1} \\ x=7 \end{gathered}$$

To find $y$,

$$\begin{gathered} y=\frac{D_y}{D} \\ y=\frac{6}{1} \\ y=6 \end{gathered}$$

The solution for the system of linear equation is $(7,6)$

Step 6. Check by substituting values

Substitute $x=7,y=6$ in both the equations,

$$\begin{gathered} x-2y=-5 \\ 7-2\left(6\right)=-5 \\ 7-12=-5 \\ -5=-5 \end{gathered}$$

This is true.

$$\begin{gathered} 2x-3y=-4 \\ 2\left(7\right)-3\left(6\right)=-4 \\ 14-18=-4 \\ -4=-4 \end{gathered}$$

This is also true.

Therefore, the solution for the system of linear equation is $(7,6)$