Q. 31 - 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.

$\left\{\begin{array}{l}3x-5y=3\\ 15x+5y=21\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system of equations$\left\{\begin{array}{l}3x-5y=3\\ 15x+5y=21\end{array}\right.$ is $\left(\frac{4}{3},\frac{1}{5}\right)$.

Step 1. Given information

The given system of equations is

$\left\{\begin{array}{l}3x-5y=3\\ 15x+5y=21\end{array}\right.$

Step 2. Determinant

The determinant D of the coefficients of the variables

$$\begin{gathered} D=\left|\begin{array}{cc}3& -5\\ 15& 5\end{array}\right| \\ =\left(3\right)\left(5\right)-(-5)\left(15\right) \\ =15+75 \\ =90 \end{gathered}$$

$D\ne 0$, so Cramer's rule can be used to determine the solution of the system of equations

Step 3. Solution for x

According to Cramer's rule $x=\frac{D_x}{D}$

Determine $D_x$

$$\begin{gathered} D_x=\left|\begin{array}{cc}3& -5\\ 21& 5\end{array}\right| \\ {D}_x=\left(3\right)\left(5\right)-(-5)\left(21\right) \\ {D}_x=15+105 \\ {D}_x=120 \end{gathered}$$

Then 

$$\begin{gathered} x=\frac{D_x}{D} \\ x=\frac{120}{90} \\ x=\frac{4}{3} \end{gathered}$$

Step 4. Solution for y

According to Cramer's rule $y=\frac{D_y}{D}$

Determine $D_y$

$$\begin{gathered} D_y=\left|\begin{array}{cc}3& 3\\ 15& 21\end{array}\right| \\ {D}_y=\left(3\right)\left(21\right)-\left(3\right)\left(15\right) \\ {D}_y=63-45 \\ {D}_y=18 \end{gathered}$$

Then 

$$\begin{gathered} y=\frac{D_y}{D} \\ y=\frac{18}{90} \\ y=\frac{1}{5} \end{gathered}$$

So the solution of the system of equation is$\left(\frac{4}{3},\frac{1}{5}\right)$