Revision:Solving Equations

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Solving Equations

Trial and Improvement

Any equation can be solved by trial and improvement (/error). However, this is a tedious procedure.

Example: Solve by trial and improvement.

Firstly, select a value of to try in the equation. I have selected . Put this value into the equation. We are trying to get the answer of 17.

If , . This is lower than 17, so we try a higher value for t.

If , (too high)

If , (too low)

If , (too high)

If , (too low)

If , (too high)

So we know that t is between 2.44 and 2.445. So to 2 decimal places, t = 2.44.

Iteration

This is a way of solving equations. An iteration formula might look like the following:

You are usually given a starting value, which is called .

If , substitute 3 into the original equation where it says . This will give you . (This is because if , and ).

(by substituting in 3).

To find , substitute the value you found for .

Repeat this until you get an answer to a suitable degree of accuracy. This may be about the 5th value for an answer correct to 3s.f. In this example,

Example

a} Show that is a rearrangement of the equation .

b) Use the iterative formula together with a starting value of to obtain a root of the equation accurate to one decimal place.

Answers

a) multiply everything by :

so

so

b)

(substitute -2 into the iteration formula)

(substitute -1.2 into the above formula)

Then:

therefore, to one decimal place, .

Comments

This article is only concerned with looking at numerical methods - it could do with being renamed.

Another article is needed looking at general methods for solving (linear) equations. A link to the article on solving quadratic equations should be include as well as links to any other relevant pages.