Guess and Check, Work Backward
Key Questions

You should use the guess and check method when you do not know how to solve a problem.
The guess and check method includes:
 make a logical guess
 test your guess
 adjust your guess based on results of #2 until you are correct
Example:
There are 20 children in the kindergarten class. The children are a mix of 5 year olds and 6 year olds. The total age of the children equals 108 years. How many 5 year olds are there?
Guess & check method:
 Let's guess that there are 10 five year olds.
 If there are 10 five year olds, there must be 10 six year olds since there are 20 children in total. Their combined age is equal to (10 x 5) + (10 x 6), or 110 years.
 Since 110 years is greater than 108 (the correct answer), our initial guess was incorrect. To get closer to the correct answer, we need to guess a higher number of five year olds (since five years is less than six years).
 Let's now guess that there are 12 five year olds.
 If there are 12 five year olds, there must be eight six year olds since there are 20 children in total. Their combined age is equal to (12 x 5) + (8 x 6), or 108 years. Therefore, the correct answer is 12 five year olds.

I am not certain the following is a fully satisfactory answer to this question, but it explains the logic behind logical transformations from something we want to prove to an obviously true statement and back.
"Working backwards" is the typical final part of a proof when from a statement we want to prove, for instance
#x^24x+6>1# , we derive an obviously true statement using some invariant (equivalent) transformations like these:
#x^24x+6 = (x2)^2+2# , which is greater or equal to 2 (since a square of any algebraic expression is greater or equal to#0# ), which, in turn, is greater than#1# .The problem with the above "proof" is that, if the initial statement was false, using seemingly correct transformations, we can come up to an obviously true statement. So, the fact that from our original statement we have derived the obviously true final statement does not necessarily prove that our initial statement was true.
But, if all transformations we made are not only "correct", but invariant (or equivalent), which, in short, means reversible, then after we have derived a true statement we can conclude:
since all transformations are invariant (that is reversible), from the final true statement we can derive the initial, and that is the actual proof.
This is actually the "working back" part of a proof.For the example above the real proof is the following sequence:
#(x2)^2>=0#  add 2 to both sides 
#(x2)^2+2>=2>1#  open parenthesis 
#x^24x+4+2>1#  simplify 
#x^24x+6>1#  which is what we had to proof. 
Answer:
The guess and check method is a form of a trial and error method.
pick a number for the equation plug it in and see how close the answer is. Adjust and guess again.Explanation:
An example is artillery fire. The gunners pick a number that is thought to be close by a little big. The place of the hit is recorded.
Then the gunners pick a number that is thought to be a little small.
The place of the it is recorded. Now the real answer is located somewhere in the middle. From the distances ( or values) above and below the actual answer can be calculated. Another guess is made usually right on or very close.
Questions
Properties of Real Numbers

Properties of Rational Numbers

Additive Inverses and Absolute Values

Addition of Integers

Addition of Rational Numbers

Subtraction of Rational Numbers

Multiplication of Rational Numbers

Mixed Numbers in Applications

Expressions and the Distributive Property

When to Use the Distributive Property

Division of Rational Numbers

Applications of Reciprocals

Square Roots and Irrational Numbers

Order of Real Numbers

Guess and Check, Work Backward