Quickly Solve Simultaneous Equations up to several variables using just a calculator (or Excel).

Often times, it's necessary to solve simultaneous equations and this can be for optimizing a voltage divider using two outer resistors and a center adjustment pot, or or countless other purposes in electronics, physics, and science. Using just a scientific calculator or a spreadsheet, it's quite easy and fast to solve a 3, 4, 5, variable or even more, simultaneous equations. Few of us would want to do this by hand as we were taught in  engineering school. Personally, I use an HP48 or 49 Scientific Calculator to solve these, but there are many brands of scientific calculators that have a Matrix Calculation capability. Specifically, the calculator needs to perform an inverse matrix multiplication, which is effectively a matrix division. I'll give an example here. 

EXAMPLE: SOLUTION TO 3-VARIABLE, SIMULTENEOUS EQUATIONS:

Here's an example set of 3 equations in 3 unknowns. For the solution to work, there needs to be at least one equation for each independent variable. 

  5*X +    9*Y + 14*Z = 31

12*X +   2*Y  -   4*Z =    8

  3*X + 16*Y  + 21*Z = 19

We first put this in standard form as follows:

|  5      9   14  |      | X |       | 31 |

| 12     2    -4  |      | Y |  =   |  8  |

|  3   16    19 |      | Z |        | 19 |

I apologize for how bad the matrix form looks above, but there are no drawing tools I could use to make a proper left and right matrix bracket, so the best I could do was to use the vertical line character.

To solve this, we simply put it into the calculator with the right-side of the equal sign first, followed by the larger matrix, then divide. When I say divide, this isn't strictly correct as there is no true division in matrix operations. It's actually doing a matrix inverse on the large matrix, then taking the smaller, 3 number column vector by the matrix using a right, matrix multiplication. On the HP48 calculator, it get's entered as follows using the matrix editor:

| 31 |

|   8 |

| 19 |

/

|    5    9  14 |

|  12   2    -4 |

|   3   16  19 |

with the result being a 3 number column matrix with the answers to the variables X, Y, and Z. Here is the result from the HP48 calculator as follows:

X = 2.48428

Y = -3.6101

Z = 3.64779

If you plug these values for X, Y, and Z back into any of the original equations, you'll find that they satisfy the equations.