Evaluation Lesson

Algebra ->  Algebra Lessons  -> Evaluation -> Evaluation Lesson      Log On


"Evaluation" mostly means "Taking care of the multiplication signs, plugging numbers in for variables, and then doing arithmetic calculations". So to do evaluations, you should be well versed in arithmetics. For those of you who did not sweat arithmetics, hopefully you have a little brother or a sister who still remembers how to do it. Just kidding. Usually, evaluation in algebra refers to an expression involving "variables", which are represented by letters like "x" or "a". They then tell you what numbers these variables represent, and you just plug these numbers instead of the variables. But dangers lurk on your path.

Okay, by now you may be impatiently waiting for examples.

Problem Solution
Expression to evaluate: 2x - 3xy
Variable x is: 5
Variable y is: -2 		Insert multiplication signs: 2*x-3*x*y
Substitute 5 for x: 2*5-3*5*y
Substitute (-2) for y: 2*5-3*5*(-2)
Do arithmetics: 10-(-30)
More arithmetics: 10+30
More arithmetics: 40
(40 is your final answer)

If you came to this site with arithmetics homework, what they want from you is to simply perform all arithmetics operations in the right order. Your problem would normally have no unknowns involved. Since this site concentrates more on algebra homework than on arithmetics, I assume that you more or less remember arithmetics and need some help with unknowns and how to properly plug them in.

Taking Care Of Multiplication Signs

Back in the good old days when you were doing arithmetics, you always knew that 52 means 52 and not 5 multiplied by 2. Now, when you study algebra, they suddenly present you with expressions like 5x and tell you that it means 5 multiplied by x. And the worst of it is that they are right. When letters (variables) are involved in expressions, you are allowed to skip the multiplication sign.

The historical reason for this is that in ancient times, ink was expensive and mathematicians were lazy, so they figured that they could save some ink and time by skipping the multiplication sign in places where it was easy to figure out that it was implied. Ink is cheap now, but the mathematicians are just as lazy, so the old tradition persisted.

Where this tradition gets you in trouble is when you need to substitute numbers for variables. If they tell you that for 5x, you should substitute 2 for x, they want you to multiply 5 by 2 (to get 10), not just write down 52 as the (incorrect) answer.

What you have to do to aviod this, ALWAYS, is to first rewrite the expression with multiplication signs explicitly put in. When you get considerable practice evaluating expressions, you may do this step implicitly in your mind, but you will always have to do it.

Expression Rewritten with multiplication signs
xyz x*y*z
xy-3(xz-y) x*y-3*(x*z-y)
%28x%2Bab%29%2F%28xy%2B1%29 %28x%2Ba%2Ab%29%2F%28x%2Ay%2B1%29
1%2F%281%2B1%2Fxyz%29 1%2F%281%2B1%2F%28x%2Ay%2Az%29%29

Where to insert multiplication signs

By now, you are most likely getting the picture. Multiplication signs are omitted in the following cases:

Expression Expanded Explanation
xy x*y When two letters go together
3y 3*y When a number goes right before a letter
3(x+1) 3*(x+1) When a number goes right before an expression in brackets
x(x+1) x*(x+1) When a letter goes right before an expression in brackets

What you have to do is insert the multiplication signs everywhere where they have been omitted. Apply the rules above until all possible multiplication signs have been put in explicitly.

Plugging Numbers

While plugging numbers is easy, there is a common pitfall in this. Be careful with negative numbers. Negative numbers should be plugged in using parentheses. Like this: (-3). Positive numbers should just be written in "as is".
Expanded Expression Variables Plugged In
x*y x=2
y=-3.5 		2*(-3.5)
x=2
y=-3 		1%2F%281%2B2%2A2%2A%28-3%29%29
(a-b)-c a=2
b=-1
c=3 		(2-(-1))-(-3)

Doing Arithmetics Calculations

After performing the steps above, you would be left with an arithmetic expression involving only numbers and operations. Just evaluate it according to the rules of arithmetics. Here are the steps in case if you forgot:

  • If the expression involves parentheses, first work on what is inside the parentheses to reduce it to just a number. Work on it as though it is an expression of its own. You should get to the point where the expression inside the parentheses is reduced to a single number. In doing so, you may use all the rules outlined here. Remember that an expression may involve nested parentheses, and you should work on eliminating them one after another, starting with the innermost ones.
  • If the expression involves powers, do the exponentiation first until there is no exponentiation left to be done. Remember that the exponent applies to the last term. For example, 5*32 means, "take 5 and multiply it by 3 squared".
    Do not forget the rules of multiplication involving negative numbers.
  • If the expression involves multiplication, do it right after you are done with powers.
  • Addition should be done last.
Expression With Numbers Plugged In Operations
3*(-5)
  • 3*(-5) = - (3*5)
  • -(3*5) = -15
3*(-3-(-5*2+8)+8)+42
  • Innermost bracket: 3*(-3-(-10+8)+8)+42
  • Innermost bracket: 3*(-3-(-10+8)+8)+42
  • Innermost bracket: 3*(-3-(-10+8)+8)+42
  • Innermost bracket: 3*(-3-(-2)+8)+42
  • Innermost bracket eliminated: 3*(-3+2+8)+42
  • Next Innermost bracket: 3*(-3+2+8)+42
  • Innermost bracket: 3*(-3+2+8)+42
  • Innermost bracket: 3*(7)+42
  • Innermost bracket eliminated: 3*7+42
  • Exponentiation (42=16): 3*7+16
  • Multiplication: 21+16
  • Addition: 37

Again, your main task is not to be in a hurry, do all the steps even if they seem tedious, and use brackets appropriately.

See Also: Evaluation Calculator