Welcome to a lesson on evaluating algebraic expressions.
An expression in which some or all of the terms contain variables is an algebraic expression or variable expression.
If the value of the variables is known, we can evaluate the expression.
So to evaluate an algebraic or variable expression, we substitute the value of the variables into the expression and then evaluate using the order of operations.
So, for example, if we have the expression
five X minus twelve and we're also given that X equals seventeen,
we replace the variable X with the value of seventeen and then evaluate the expression.
So here we'd have five times seventeen minus twelve.
Well five times seventeen is eighty-five and then eighty-five minus twelve is equal to seventy-three.
Let's go ahead and review the order of operations before we do a few more examples.
So remember, there are four steps to the order of operations.
Number one, we'll evaluate inside the parentheses or grouping.
Number two, we'll evaluate the exponents.
Number three, we'll multiply and divide from left to right as it occurs.
And number four, we'll add and subtract from left to right.
So let's go ahead and look at several examples.
So here we're given that X equals negative three, Y equals seven, and Z equals negative two.
And we want to evaluate X minus three Y plus seven.
Well X is equal to negative three, so we'll have negative three plus three times Y, but Y is seven, plus seven.
Now we'll follow the order of operations so we'll multiply first.
Three times seven would be twenty-one so we have negative three plus twenty-one plus seven.
And now we just add from left to right so negative three plus twenty-one, that would be eighteen. Plus seven equals twenty-five.
This next example is the same process. We have two times X squared, but X is now negative three.
Plus five times Y, Y is seven.
Minus Z to the third power and Z is equal to negative two.
And so now, we'll evaluate this using the order of operations.
There's nothing we can do inside the parentheses, so next we'll take a look at the exponents.
Negative three squared would be positive nine.
And negative two raised to the third power would be equal to negative eight.
Now, we'll go ahead and multiply. So two times nine is equal to eighteen.
Plus, five times seven is equal to thirty-five.
And here we have minus negative eight and that'll become plus positive eight.
And now, performing the addition from left to right
we have eighteen plus thirty-five plus eight.
Eighteen plus thirty-five would be fifty-three plus eight would be sixty-one.
Now we have Y equals positive two and we want to evaluate this expression here. So we'll replace Y with positive two.
Next, there's no parentheses that we can simplify, there's no exponents, so we will multiply and divide now from left to right.
Now here we have a fraction. This does mean nine divided by two.
So it really depends if we want to have our final answer in fraction form or decimal form.
Let's go ahead and rewrite this in decimal form.
So nine divided by two would be four point five.
Minus eight times two, that's sixteen plus two.
And now we'll perform the addition and subtraction from left to right.
So we have four point five minus sixteen, that's going to be negative eleven point five.
Plus two, which equals negative nine point five.
If we wanted this in fraction form, that would be the same as negative nine and one-half.
Or we could write this as negative nineteen-halves. All of which are equivalent, just in different forms.
Here we have X equals three and Y equals negative five, so we'll have four times X which is three minus three time Y squared but Y is negative five.
So we'll have negative five squared.
Next we'll evaluate negative five squared.
That'll be four times three minus three times negative five squared is positive twenty-five.
Now we'll multiply. So three times four would be twelve minus three times twenty-five is equal to seventy-five.
And then, finally, twelve minus seventy-five is equal to negative sixty-three.
And let's try another.
Here we have X equals negative two, so we'll substitute negative two in for X.
So we'll have three times negative two to the third power minus negative two squared plus two times negative two plus nine.
Next, we'll simplify the exponents.
Negative two raised to the third power is equal to negative eight.
Negative two squared would be positive four.
And now we'll multiply. Three times negative eight equals negative twenty four.
Here we'll have minus four plus two times negative two, that's negative four. And then plus nine.
So now we'll add and subtract from left to right.
So negative twenty-four minus four, that'll be negative twenty-eight.
Then negative twenty-eight plus negative four, that'll be negative thirty-two.
And then, lastly, negative thirty-two plus nine equals negative twenty-three.
Let's go and take a look at one more example.
Here we have X equals two and Y equals negative three.
So the numerator will be two squared times negative three squared.
Over X squared, which would be two squared, minus two times Y to the third will become negative three to the third.
So what we'll do here is evaluate the numerator and denominator separately, and then lastly, simplify the fraction.
So two squared would be four, and negative three squared would be nine.
In our denominator we're gonna have four minus two times negative three cubed is equal to negative twenty-seven.
Our numerator is going to be thirty-six.
In the denominator, we'll have four minus negative fifty-four, which will become plus fifty-four.
So this fraction is thirty-six over fifty-eight. These both have a common factor of two.
So in simplified form, the fraction would be eighteen over twenty-nine.
I hope you found this video helpful, and thank you for watching.