Lesson 3.3 Equivalent Expressions

Integrated Lesson 3.3: Proving if Equations are Equivalent

This lesson will begin with a brief video demonstrating how algebraically equations can be proven equivalent. I will then provide example problems to follow and then some open-ended problems at the end that will be your class assignment

​You may use graphing to check whether equations are equivalent but not to prove this as instructed algebraically! As well, substituting values in can demonstrate some equivalency but does not prove the equations for all values of x

• Mrs. Hatch

Again, you can use graphing as a back-up but it will not be accepted for full points! The intention is for you to use algebra process to prove equivalency or to show whether the statement is an identity 

​I will now go through 3 example, so please take some notes for this lesson :)

​Example #1: Prove that x³ - y³ = (x - y)(x² + xy + y² )


**The simplest approach is to multiply wherever the expression includes multiplication!

In your notes, multiply the right side using distributive property or the table method and then answer the next question regarding the 6 items you should have after the multiplication step

Multiple Choice

What are the 6 terms you get after multiplying (x - y)(x² + xy + y² )?

• x³ + x²y + xy² - x²y - xy² - y³

• x² + x²y + xy² - x²y - xy² - y²

• x³ + x²y + xy² + x²y + xy² - y³

The six items when written out are
x³ + x²y + xy² - x²y - xy² - y³

​I used the table method

​Notice the green items are like terms and the yellow items are like terms....on the next slide, what will the cleaned up version be after combining like terms?

​The like terms cancel each other out so we are left with x³ - y³ which is exactly what the left side of the initial equation was. This proves that the left is equivalent to the right.



By the way, QED is an acronym for "Quad erat desideratum" which is Latin for "that which was desired." I like to finish math proofs with this acronym because I seem fancy ;)

​Example #2: Prove that x⁴ - y⁴ = (x - y)(x² + y² )(x + y)


**The right side has three things to multiply so try to choose two that you think would be easiest. The (x-y) and (x+y) are conjugates of each other so when you multiply them, the middle terms will cancel each other out, so let's multiply those together first.

In your notes, multiply (x - y)(x + y) using distributive property or the table method and then answer the next question regarding the 4 items you should have after the multiplication step

Multiple Choice

What are the 4 terms you get after multiplying (x - y)(x + y)?

• x² + xy + xy - y²

• x² + xy - xy - y²

• x² - xy - xy - y²

The four items when written out are
shown in red to the right of the set-up; the middle terms cancel each other out!

​I used distributive (FOIL)

Now we must multiply this term by the (x² + y²)

Do this and go to the next slide to see the completion of this problem

​The next two binomials are also conjugates (same items but one + and one -) so the middle terms dropped out. The right side matches the left so the expressions are equivalent.



And another fancy QED pointing that the left = the right

​Example #3: Determine if the equation is an identity
(x + 3)(x - 1)² = (x² - 2x - 3)(x - 1)


While the directions are different, this has the same process to try and match the left with the right.

If the left IS NOT equivalent to the right, then this is not an identity

If the left IS equivalent to the right, then it is an identity



Notice that both sides have (x - 1) but the one on the left has an exponent. Write the one on the left expanded to see what we have:

​(x - 1)² is the same as (x - 1)(x - 1) so that is what I meant by expanding the left side!

Now you can see that they both have a (x - 1) which are underlined in yellow. Do the other parts match?

Multiply the left binomials to see if they match the trinomial on the right

Multiple Choice

When the binomials on the left are multiplied, do they match the trinomial on the right?

• Yes!

• Nope!

​Left does not equal right, thus Not an Identity

Now do the Class Assignment. All of them have the same instructions which are to determine whether they are an identity. Do the problems on paper and then submit your answers on the next three slides.

When finished, turn in the work on paper :)

Multiple Choice

#1 in Class Assignment: Is this equation an identity?

• Yes!

• Nope!

Multiple Choice

#2 in Class Assignment: Is this equation an identity?

• Yes!

• Nope!

#3 has a LOT of multiplying to do........Expand the binomials that have exponents and then start multiplying and looking for things on the left that could match those on the right.

Multiple Choice

#3 in Class Assignment: Is this equation an identity?

• Yes!

• Nope!

Be sure to turn in your work on paper for the three class assignment problems for credit!