**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__

- Estimate two consecutive numbers that will possibly be a solution.
- Substitute and solve using the equation.
- 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__

- Rearrange the equation.
- 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.