Welcome to a lesson
on the difference between an expression and an equation.
An expression is a mathematical phrase
that combines numbers and/or variables
using mathematical operations.
An expression can be evaluated but not solved,
and therefore expressions do not contain equal signs.
Here are three examples of expressions.
Notice how we can evaluate this first expression,
since two plus two equals four,
but for the second and third expressions,
we would have to be given a value of x
to evaluate the expression.
So we'll come back to this,
but let's define an equation.
An equation is a mathematical sentence
that combines numbers and/or variables,
that shows the equality of two expressions.
So the key here is an equation shows equality.
And therefore an equation is two expressions
set equal to each other,
and normally we can solve equations,
but sometimes equations don't have solutions.
But the key to remember here is that because an equation
expresses equality, all equations must contain equal signs.
Looking at our three examples,
notice how all three of them contain equal signs
showing two expressions equal to each other.
So before we look at the solutions to these equations,
let's go back up to our expressions.
Again, we already said that we know that two plus two
is equal to four, and therefore we can write
an equation to show the value of this expression.
I think this is where some of the confusion
comes into play.
Again, we know that two plus two equals four,
so we just wrote an equation to show
the value of the expression two plus two.
But again, we were given an expression, not an equation.
For the expression three x minus five,
let's say we were given x equals ten.
Well, if we're given a value of x,
we can evaluate this expression in this case
by substituting ten for x.
That would be three times ten instead of three times x
minus five, this would be 30 minus five or 25.
Again, notice how we just wrote an equation
to show that the expression is equal to 25
when x equals 10.
But again, we were given an expression, not an equation.
So this is not a solution, this is just the value
of the expression when x is 10.
For our third expression,
let's say we were given x equals five,
so this is not a solution,
this is just a given value of x.
So we'll substitute five for x to evaluate the expression.
Well this would be 25 minus 10, that's 15,
minus eight, which is equal to seven.
Again, we just wrote an equation to show
the value of the expression when x equals five.
Now let's go back down and look at our equations.
Two plus two equals four is an equation,
even though it doesn't contain a variable,
notice how this equation is always true,
and therefore has a special name,
it's called an identity.
Next, the equation three x minus five equals seven
has a solution.
The solution is x equals four.
The reason it's a solution, is that when x equals four,
the expression on the left is equal to the expression
on the right, or if we substitute four for x,
we would have three times four minus five equals seven.
Well this would be 12 minus five, which is seven.
Seven equals seven verifies x equals four is our solution.
And for our last equation, because of the x squared
we actually have two solutions.
One solution is x equals four,
and the other solution is x equals negative two.
Which means if we substitute four for x,
or negative two for x, the expression on the left
will equal the expression on the right,
or the left side will equal zero.
Let's go ahead and verify that.
If x is four we would have four squared,
minus two times four minus eight equals zero.
Well, this would be 16 minus eight minus eight,
which does equal zero, so that's true,
x equals four is a solution,
and then when x equals negative two,
we would have negative two squared,
minus two times negative two minus eight equals zero.
We have four plus four minus eight which does equal zero,
verifying our solutions.
So again, the main thing to remember is an expression
does not contain an equal sign,
therefore it can't be solved, but it can be evaluated,
and an equation must contain an equal sign,
and in many cases can be solved.
We'll just emphasize this one more time,
determine which of these are expressions
and which are equations.
Let's go ahead and circle the expressions
and put boxes around the equations.
First we have x cubed equals eight.
We have an equal sign, and therefore we have an equation.
Next we have two x to the fourth
minus five x squared plus three.
No equal sign, therefore this is an expression.
Next we have the square root
of the quantity x minus two.
No equal sign, therefore we have an expression.
Next we have c equals two pi r.
We have an equal sign, and therefore an equation.
This equations happens to be the circumference of a circle.
Next we have pi r squared, no equal sign,
therefore it's not an equation, it's an expression.
And then finally, we have x equals negative b
plus or minus the square root of the quantity
b squared minus four a c, all divided by two a.
We have an equal sign, and therefore we have an equation.
This is actually a very famous formula,
called the quadratic formula.
Okay, I hope this video helps clarify the difference
between an expression and an equation.
Thank you for watching.