What is evaluation of algebraic expressions? How do we do it?

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Algebraic expressions are the mathematical relations explaining a general trend in a changing activity. For example, consider adding two pound of sugar to a bucket of sugar which already contains some sugar in it. 

Do we know how much sugar is already in the bucket? No.

Do we know how much more sugar got added? Yes.

How much sugar in the bucket now? We don't know. But we do know one fact with certainty that now the sugar is 2 pounds more.

So the unknown is "how much sugar was there in the bucket already". Isn't it? Let's use a lower case letter to represent this unknown.

Consider the sugar already in the bucket = s 

Now we have added another 2 pounds of sugar to "s" so now we have a very very simple algebraic expression "s + 2" which represent the amount of sugar in the bucket after adding 2 pounds more into it.

Let's say the person who put the sugar into the bucket before he/she came and told you that he/she put 5 pounds of sugar into the bucket earlier. 

Isn't it the value of "s"?

Yes it is. If so, then we can replace "s" by "5" in our expression as shown below:

"5 + 2" 

Should be leave it there or do we need to do something? 

We know what "5 + 2" is, don't we? YES we do know. Then solve it and write the answer as shown below:

5 + 2 = 7

The above procedure is called evaluating an algebraic expression.

In other words, once we revealed the number value of a variable then we can substitute that value for the variable in the given algebraic expression and solve it, which is called the evaluation of the algebraic expressions.

Below are some more examples in pure math language, hopefully will make sense from the above explanations:

Consider the simple word phrase “twice a number added to 8”. The algebraic expression for this phrase is given below; 

2n + 8 

Where “n” represents any number, two is its coefficient (as twice means 2 times) and 8 is the constant term. Now “n” represent any number, which means, “n” can take any value from the countless numbers in the real number system. 

Let’s evaluate this expression by giving some number values to “n”; 

For example, when the number is 3 then the expression is evaluated as follows; 

2(3) + 8 = 6 + 8  = 14 

Note that, instead writing 2n, I wrote 2(3), by substituting “3” for “n”. 

Always use parenthesis when substituting  the variable values in an algebraic expressions. 

Now let's do another example for  n = - 2, the algebraic expression is evaluated as follows; 

2(- 2) + 8  = -4+ 8 = 4  

Summary:  In an algebraic expression, a variable or unknown can take many values depending upon the circumstances. When we substitute a particular number value for a variable and  solve the algebraic expression according to the order of operations, this procedure is known as the evaluating the algebraic expression.