Saturday, 16 August 2008

Graphical Solution of Equations

There are many ways to solve a mathematical equation.

However, there are some equations that would be harder to solve through the conventional mathematical approach using, for example, factorisation, quadratic formula or completing-the-square method.

To solve equation like 1 - x + 6 sin x = 0 would be almost impossible with the above stated methods.

Graphical technique to solve these sort of maths equations will be the better option.

How is the graphical technique applied ?

We will use the above equation 1 - x + 6 sin x = 0 to explain.

To simplify the drawing of the stated equation graphically, we can rearrange the expression to be,

1 - x + 6 sin x = 0 ==> 6 sin x = x - 1.

Let y = 6 sin x, and therefore, y = x - 1 also.

The original equation is divided into 2 parts with the introduction of "y".

We can now sketch, with ease, the 2 graphs y = 6 sin x and y = x - 1 onto the same sheet. The purpose of doing this is similar to solving simultaneous equations.

The sketch of the 2 individual graphs is shown below.

y = 6 sin x overlaps with y = x - 1 at 3 different points.

These 3 points will be the roots (or solutions) of the expression 1 - x + 6 sin x = 0.

Answers: x = -2.5, -0.2 and 2.8

Though these 3 answers may be estimated graphically, the method used to arrive at them is obviously simple.

This is the power of graph and its solution of not so-easily-solved maths equations.

Through this post, you will start to get the feeling that maths is indeed interesting (if you have not yet like math).


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