Question

The system of equations is given:

2x − 4y + 6z = 14
9x − 3y + z = 10
5x + 9z = 1
a. Solve the system using algebra or matrix operations. If you use matrix operations, include the matrices you
entered into the software and the calculations you performed to solve the system.
b. Verify your solution is correct.
c. Justify your decision to use the method you selected to solve the system.

Answer 1

x=-1/85; y=-283/85; z=2/17

Explanation:

Using an algebraic method like elimination or substitution would take a lot of steps which could lead to mistake the calculations. In this case, I decided to use the Gaussian elimination. We can express the system in matrix form as follows:

`\left[\begin{array}{ccc}2&-4&6\\9&-3&1\\5&0&9\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}14\\10\\1\end{array}\right]`

To begin the calculations, we write the system in augmented matrix form and use the Gaussian elimination:

`\left[\begin{array}{ccccc}2&-4&6&|&14\\9&-3&1&|&10\\5&0&9&|&1\end{array}\right]`

By applying the Gaussian elimination, the final matrix is the following:

`\left[\begin{array}{ccccc}1&0&0&|&-1/85\\0&1&0&|&-283/85\\0&0&1&|&2/17\end{array}\right]`

In order to verify the results, it´s enough to substitute the calculated values in the original equations to see if the equalities are correct. Here you can see the verification for all of the equations:

`2(-1/85)-4(-283/85)+6(2/17)=14\\9(-1/85)-3(-283/85)+1(2/17)=10\\5(-1/85)+9(2/17)=1`