Q. 22 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}2x+4y=16\\ 3x-5y=-9\end{array}\right.$

Step-by-Step Solution

Answer

The solution is $x=2, \mathrm{y}=3 \mathrm{and} \mathrm{in} \mathrm{ordered} \mathrm{pair} \mathrm{is} \left(2,3\right).$

Step 1. Given Information

The given system of equation is

$\left\{\begin{array}{l}2x+4y=16\\ 3x-5y=-9\end{array}\right.$

We have to solve the system of equations using Cramer's rule if it is applicable.

Step 2. Finding the determinant

The determinant D of the coefficients of the variables is

$$\begin{gathered} D=\left|\begin{array}{cc}2& 4\\ 3& -5\end{array}\right| \\ D=\left(2\right)\left(-5\right)-\left(4\right)\left(3\right) \\ D=\left(-10\right)-\left(12\right) \\ D=-22 \end{gathered}$$

Since $D\ne 0$ Cramer's rule is applicable.

Step 3. Use Cramer's rule

Let's find $D_x \mathrm{and} {\mathrm{D}}_{\mathrm{y}}$

$$\begin{gathered} D_x=\left|\begin{array}{cc}16& 4\\ -9& -5\end{array}\right| \\ {D}_x=\left(16\right)\left(-5\right)-\left(4\right)\left(-9\right) \\ {D}_x=-80+36 \\ {D}_x=-44 \end{gathered}$$

And

$$\begin{gathered} D_y=\left|\begin{array}{cc}2& 16\\ 3& -9\end{array}\right| \\ {D}_y=\left(2\right)\left(-9\right)-\left(16\right)\left(3\right) \\ {D}_y=-18-48 \\ {D}_y=-66 \end{gathered}$$

Step 4. Finding x and y

By using Cramer's rule, the values of x and y are 

$$\begin{gathered} x=\frac{D_x}{D} and y=\frac{D_y}{D} \\ x=\frac{-44}{-22} and y=\frac{-66}{-22} \\ x=2 and y=3 \end{gathered}$$