Solving Equations – Trial, Improvement & Iteration

Solving Equations

There are many ways to find the value of a variable. However, let’s study solving for the unknown using trial, improvement, and iteration.

Trial and Improvement

This is a method of finding the value of an unknown by estimating the possible value of the variables and trying to solve it using the equations.

Steps in solving Trial and Improvement

1. Estimate two consecutive numbers that will possibly be a solution.
2. Substitute and solve using the equation.
3. Determine if the estimated value is too high or too low, then improve the estimate to be nearer the solution.

Let’s have this example: x^2 + 2x = 50
The two estimated consecutive numbers for this example will be 6 and 7.
x^2 + 2x = 50
x = 6
(6)^2 + 2(6) = 50
36 + 12 = 50
48 = 50               Too low
Since 50 is higher than 48. Proceed to the second estimate, which is 7.
x = 7
(7)^2 + 2(7) = 50
49 + 14 = 50
63 = 50                 Too high
63 is higher than 50. So the value of x is between 6 and 7. Let’s try 6.1 and 6.2 because if x=6 is slightly closer to 50 than x = 7.
x = 6.1
(6.1)^2 + 2(6.1) = 50
37.21 + 12.2 = 50
49.41 = 50             Too low

x = 6.2
(6.2)^2 + 2(6.2) = 50
38.44 + 12.4 = 50
50.84 = 50            Too big

We can still continue to two decimal places. Try 6.14 and 6.15.
x= 6.14
(6.14)^2 + 2(6.14) = 50
37.7 + 12.28 = 50
49.98 = 50             Too low

x = 6.15
(6.15)^2 + 2(6.15) = 50
37.82 + 12.30 = 50
50.12 = 50             Too big

Therefore the solution for the value of x is between 6.14 and 6.15.

Iteration

Solving using the iteration method is rearranging the equation to solve the equation. The starting of the x sub 0. After solving, it will lead to x sub 1 and continue doing it until the proper value.

Steps in solving iteration

1. Rearrange the equation.
2. Use the iterate equation then solve.

For example x^2 – x – 6 = 0,                        This is the possible equation.
[1] x = 6 – x^2                                                Add both sides x to get the first equation.
[2] x^2 = x + 6
         x = √(x+6)                                             Add (x + 6) on both sides then the root.
[3]x=  (x+12)/x

Use iterative formula x=  (x+6)/x with a starting point of 4.
x = (4 + 6)/4 = 2.5
x = (2.5 + 6)/2.5= 3.4
x = (3.4 + 6)/3.4 = 2.76
x = (2.76 + 6)/2.76 = 3.17
 x = (3.17 + 6)/3.17 = 2.89
x = (2.89 + 6)/2.89 = 3.07

Therefore the value is between 2.89 and 3.1.