How do i solve simultaneous equation with more than two equations and two unknowns?

While simultaneous equation with two equations and two unknowns should be easy enough to solve using elimination, substitution, or graph sketching, simultaneous equations with more than two equations and two unknowns is much trickier and more tedious. Firstly we need to make sure the number of unknowns and the number of equation is the same. For example, if we have three unknowns, we must also have three different equations to solve the problem. Secondly, we need to make sure that all the unknowns are linear, such that the power of all unknowns are 1 and there is no non-linear functions such as sin or cos.

Method: Inverse matrix

Let's say we have three equations,

2x+6y+3z=5

4x+7y+7z=9

7x+2y+6z=3

We will need three matrix which are combination of all the three equations. The first one is matrix A, a 3X3 matrix consists of coefficient of x,y, and z from the three equations. The first row is from the first equation, the second row is from the second equation, while the third row is from the third equation. The first column is the coefficient of x, the second column is the coefficient of y, while the third column is the coefficient of z. The second matrix is matrix X, a 3X1 matrix consists of unknowns, such that X=[ x y z ]^T . The third matris is matrix B, a 3X1 matrix consists of numbers at the right hand side of the equation. Just like matrix A, in matrix B first row is from the first equation, the second row is from the second equation, while the third row is from the third equation. Then we have a matrix equation, AX=B, we can solve it by pre-multiplying both side of the equation by A^-1.

Hence it become X=A^-1B. Do a normal matrix multiplication and you will get the answer.