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

kkazem

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.

  • in reply to javagoza

    That's interesting. I had no idea about hacking the HP48. And wow, hand inverting a12x12 matrix, that must have taken some time. Just computing the determinant would take a while, not to mention the rest of it. 

  • in reply to kkazem

    I agree that it is important to know and understand the principles of algebra and solving algorithms. That has allowed me to better understand the principles of Machine Learning.

    I remember my first algebra exam in Industrial Electrical Engineering in the early 1980s. One of the exercises was to hand invert a 12x12 matrix. The exam consisted of 2 parts: a theoretical exam in the morning of 5 hours and another exam of practical problems of 5 hours in the afternoon, a total of 10 hours of examination and an hour for lunch. We started the course 1500 students, we finished just 200, in the sixth year of the college career there were only 24 students left. That exercise was a student filter. Now I have a daughter studying the same career and fortunately things have changed. Few young people want to spend 6 years studying something so hard and have had to lower the filter.

    I also remember how we hacked the HP48 when they appeared on the market to increase the range of infrared communications that was limited as standard.Innocent

    They allowed us in some tests to use the HP48 but after the teacher performed a complete memory reset and removed any expansion modules.

  • in reply to javagoza

    To be clear, I posted this blog to help engineers and engineering students to save time when they come across simultaneous equations. In school, I completely agree, the point is to learn how matrix algebra works and to understand the basic principles like the difference between left and right side multiplication and that the results are different. 

    More important, the more modern calculators like the HP48 can be used as general purpose cheating machines by storing loads of theory and answers in the relatively huge memory space available. But my blog wasn't about that.

  • in reply to javagoza

    Not just Spain, even in Indian colleges up to 12th standard and entrance exams calculators or any kind are not allowed. However, once you enter engineering college, you are allowed to use only a non-programmable scientific calculator.  The Casio FX991-ES is the most popular calculator among Indian engineers.

  • This is one of the absurd and stupid reasons why programmable calculators and / or with matrix functions are not allowed in Spain in college entrance exams.