Question

Use cramers Rule to solve the following system:

5x - 3y + z = 6
2y - 3z = 11
7x + 10y = -13

Answer 1

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

Explanation:

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

`x= (D_x)/(D)`,
`y= (D_y)/(D)` and
`z= (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]* \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= (D_x)/(D)=(199)/(199)\\ x=1`

`x= (D_y)/(D)=(-398)/(199)\\ y=-2`

`x= (D_z)/(D)=(-995)/(199)\\ x=-5`