Question

Use cramers Rule to solve the following system:

Answer 1

Answer:

The solution to the system is $x=1$,$y=-2$ and $z=-5$

Step-by-step explanation:

Cramer's rule defines the solution of a system of equations in the following way:

$x= \frac{D_x}{D}$, $y= \frac{D_y}{D}$ and $z= \frac{D_z}{D}$ where $D_x$, $D_y$ and $D_z$ are the determinants formed by replacing the x,y and z-column values with the answer-column values respectively. $D$ is the determinant of the system. Let's see how this rule applies to this system.

The system can be written in matrix form like:

$\left[\begin{array}{ccc}5&-3&1\\0&2&-3\\7&10&0\end{array}\right]\times \left[\begin{array}{c}x&y&z\end{array}\right] = \left[\begin{array}{c}6&11&-13\end{array}\right]$

Then each of the previous determinants are given by:

$D_x = \left|\begin{array}{ccc}6&-3&1\\11&2&-3\\-13&10&0\end{array}\right|=199$ Notice how the x-column has been substituted with the answer-column one.

$D_y = \left|\begin{array}{ccc}5&6&1\\0&11&-3\\7&-13&0\end{array}\right|=-398$ Notice how the y-column has been substituted with the answer-column one.

$D_z = \left|\begin{array}{ccc}5&-3&6\\0&2&11\\7&10&-13\end{array}\right|=-995$

Then, substituting the values:

$x= \frac{D_x}{D}=\frac{199}{199}\\ x=1$

$x= \frac{D_y}{D}=\frac{-398}{199}\\ y=-2$

$x= \frac{D_z}{D}=\frac{-995}{199}\\ x=-5$