\[ \]

Simultaneous Equations

Issue No 40, 21 August 2023

By: Anthony O. Ives

Simultaneous equations are encountered when you have a number of unknown properties for which there is at least the same number of conditions hence you have the same number number of equations as unknowns. For example if you have two unknowns you will need at least two equations to work them out, if its three unknowns then you will need at least three equations to work them out, and etc. Simultaneous equations are often the principle on which a lot of software is based that uses numerical methods such as FEA (Finite Element Analysis), CFD (Computational Fluid Dynamics), etc. Determinents, matrices are also methods by which a computer can be used to solve large number of simultaneous equations for large number of unknowns in numerical methods [1]. However determinents and matrices will be looked at in future articles.

Simple simultaneous equations can be used to solve equilibrium problems [2] where you can determine a number of equations from certain conditions which will help then determine the forces in the equilibrium scenario. We start by looking at two equations to determine two unknowns and then looking at three equations to determine three unknowns as in the following examples.

Equation (1): \(2x + 3y = 4\)

Equation (2): \(x + 6y = 2\)

The principle to solving the simultaneous is you can cancel out one of the unknowns by multiplying or dividing the entire equation, for example multiplying equation (2) by 2 which would give a new equation (3) as following:

2 × Equation (2) = Equation (3): \(2x + 12y = 4\)

Then if you take equation (1) away from equation (3) the unknown value 'x' will cancel allowing you to determine the unknown 'y' as following:

Equation (3) - Equation (1): \((2-2)x + (12-3)y = (4-4)\)

Equation (3) - Equation (1): \(9y = 0\) therefore y=0

Now that 'y' has been determined you can now work out 'x' from either equation (1) or (2) as following:

Equation (1): \(2x + 3y = 4\)

Equation (1): \(2x = 4\) therefore \(x = 2\)

Equation (2): \(x + 6y = 2\)

Equation (2): \(x = 2\)

We will do another example using three equations to solve three unknowns as following:

Equation (1): \(2x + 3y + 5z= 4\)

Equation (2): \(x + 6y + 3z = 2\)

Equation (3): \(4x + 5y + 7z = 6\)

If we multiply equation (2) by 2 and take equation (1) away from we will cancel out 'x' in equation (1) as following:

2 × Equation (2) = Equation (4): \(2x + 12y + 6z = 4\)

Equation (4) - Equation (1) = Equation (5): \(9y + z = 0\)

If we multiply equation (2) by 4 and take equation (3) away from we will cancel out 'x' in equation (2) as following:

4 × Equation (2) = Equation (6): \(4x + 24y + 12z = 8\)

Equation (6) - Equation (3) = Equation (7): \(19y + 5z = 2\)

Now we have two equations with just two unknowns which we will solve as we did with the first example:

Equation (5): \(9y + z = 0\)

Equation (7): \(19y + 5z = 2\)

5 × Equation (5) = Equation (8): \(45y + 5z = 0\)

Equation (8) - Equation (7): \(26y = -2\) therefore \(y = -\frac{2}{26} = -\frac{1}{13}\)

Equation (5): \(9y + z = 0\)

Equation (5): \(9\left(-\frac{1}{13}\right) + z = 0\) therefore \(z = \frac{9}{13} \)

Equation (7): \(19y + 5z = 2\)

Equation (7): \(19\left(-\frac{1}{13}\right) + 5z = 2\) therefore \(z = \frac{45}{65} = \frac{9}{13}\)

Now you can use Equation (1), (2) or (3) to work out the remaining unknown 'x' as following:

Equation (1): \(2x + 3y + 5z= 4\)

Equation (1): \(2x - 3\left(\frac{1}{13}\right) + 5\left(\frac{9}{13}\right)= 4\) therefore \(2x = 4 + 3\left(\frac{1}{13}\right) - 5\left(\frac{9}{13}\right)\)

Equation (1): \(x = 2 + \frac{3}{26} - \frac{45}{26} = -\frac{10}{26} = \frac{5}{13}\)

Equation (2): \(x + 6y + 3z = 2\)

Equation (2): \(x - 6\left(\frac{1}{13}\right) + 3\left(\frac{9}{13}\right) = 2\) therefore \(x = 2 + 6\left(\frac{1}{13}\right) - 3\left(\frac{9}{13}\right)\)

Equation (2): \(x = 2 + \frac{6}{13} - \frac{27}{13} = \frac{5}{13}\)

Equation (3): \(4x + 5y + 7z = 6\)

Equation (3): \(4x - 5\left(\frac{1}{13}\right) + 7\left(\frac{9}{13}\right) = 6\) therefore \(4x = 6 + 5\left(\frac{1}{13}\right) - 7\left(\frac{9}{13}\right)\)

Equation (3): \(x = \frac{78}{52} + 5\left(\frac{1}{52}\right) - 7\left(\frac{9}{52}\right) = \frac{20}{52} = \frac{5}{13}\)

Going beyond three unknowns becomes more timeconsuming and tedious, this is where a computer program will become useful and in addition using matrices or determinents to solve the equations for you. However, future articles will show you how to solve the equations using either matrices or determinents [1]. This article will help you solve simpler simultaneous equation where you just need to find out two or three unknowns typcially when you are trying to work out forces in an equililbrium problem [2].

Please leave a comment on my facebook page or via email and let me know if you found this blog article useful and if you would like to see more on this topic. Most of my blog articles are on:

Mathematics
Helicopters
VTOL UAVs (RC Helicopters)
Sailing and Sailboat Design

If there is one or more of these topics that you are specifically interested in please also let me know in your comments this will help me to write blog articles that are more helpful.

References:

[1] Engineering Mathematics, K. A. Stroud, Fourth Edition, 1995, Macmillan

[2] http://www.eiteog.com/EiteogBLOG/No25EiteogBlogNewton.html

Disclaimer: Eiteog makes every effort to provide information which is as accurate as possible. Eiteog will not be responsible for any liability, loss or risk incurred as a result of the use and application of information on its website or in its products. None of the information on Eiteog's website or in its products supersedes any information contained in documents or procedures issued by relevant aviation authorities, manufacturers, flight schools or the operators of aircraft, UAVs.