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Solving Equations By Iteration | Algebra | Maths | FuseSchool

Sometimes we may not know how to solve an equation. We could use trial and improvement to find an approximate solution… however this is really
slow and tedious. Another option is iteration.

Iteration is a way of solving equations, by finding a numerical approximation of the solutions. An iterative procedure is one which is repeated many times. Each time it runs, the output becomes the input for the next cycle, and we keep going until the output matches the input. So, if we are trying to find the solution for this equation, to three decimal places.

Choose a starting value… so x is 5, and put that into the cycle. We get x is 5.916 etc This does’t equal our starting value of 5. so, then 5.916 goes back into the cycle. In goes 5.916 Out comes 5.679 which is different, so we put it back into the cycle and so the cycle keeps on going, until
eventually we get an approximate answer that is accurate enough. Rounded to 2 decimal places, 5.73 went in and 5.73 came out. The answer is 5.73!

So that’s how iteration works, itis a repeated cycle… Although the question was this formula… We used a rearranged version in our flow
diagram. This rearranged version is called the iteration formula. You are usually given the iteration formula in the question, but if you are not it’s quite simple. You simply need to rearrange the equation to make the highest power of x the subject We then repeatedly use this iteration formula to find the solution. Let’s have a look at another example.

In part a) we have to derive the iteration formula from the starting equation. This just means that we need to rearrange the start formula to
become x equals Make x-squared the subject first. We could square root everything but this is not what the question wants… so instead this time we’re going to divide everything by x. That is the iteration formula So part b, we use this formula and 3 as the starting value...

Because the question says “to 3 significant figures” we need to keep 4 significant figures each time. So, 1 degree of accuracy more than requested. 3 into our iteration formula gives 2 point 333... This is not the same as 3, so we now put 2.333 into the formula and keep going until the input and output are the same to 3 significant figures. Rounded to 3 significant figures, 2.41 goes in and 2.41 comes out. This is the final answer!

That’s all there is to it! Iteration is just another way of solving an equation, using a repeated cycle. You just keep going until your input value matches the output value. If you have any questions, comment below and ill help you out. Please ‘like’ and share our videos with your friends. Visit us at fuseSchool.org for more videos, and more teacher help.


CREDITS
Animation & Design: Jean-Pierre Louw @ www.Behance.net/Appel718
Narration: Lucy Billings
Script: Lucy Billings


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