# Solving Systems of Linear Equations One Step at a Time

In our previous post about expanding Step-by-step solutions, we introduced a revamped equation solver. I’m proud to say that it has now been extended to solve systems of linear equations. In addition, you have four different methods to choose from when looking for a solution! These methods are elimination, substitution, Gaussian elimination, and Cramer’s rule. Let’s look at *x* + *y* = 5, *x* – *y* = 1 to see all four methods in action.

First we look at elimination:

Now we solve using substitution:

Expressing our system in matrix form, we can perform Gaussian elimination:

Lastly, the determinant can play a fundamental role in solving linear systems. This is called Cramer’s rule:

The number of unknowns is not a problem for Wolfram|Alpha; here *w* + *x* + *y* + *z* = 1, 2*w* – *x* + *y* – *z* = 0, *w* + 3*x* + *y* + *z* = 5, *w* + *x* – *y* + 2*z* = 2 is an example of a 4×4 linear system. We have even added this new functionality when solving word problems.

Having various methods to choose from is a great way to look at a problem from all angles. Stay tuned for more and more step-by-step math in the future!

How to solve this system?

3x-2y+z=1;

x+3y-z=2;

x-8y+3z=-3;

Or this one?

3x-2y+z-2w-u=3,

-9x+6y-3z+6w+10u=-2,

3x-y+3z-w-u=4,

Above systems have parametric solutions. But wolfram don’t say which variables are parametric and which are static.

This systems you can solve with matrix, so step by step solution by Gauss elimination would be nice.

We don’t have steps for underdetermined or overdetermined systems at the moment, but we can treat the second equation as determined with the following input: http://www.wolframalpha.com/input/?i=solve+3x-2y%2Bz%3D1%3B+x%2B3y-z%3D2%3B+x-8y%2B3z%3D-3+for+x%2Cy%2Cz