Rewriting Algebraic Expressions Using Structure

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Algebraic expressions can be written in more than one way. In this lesson, we will go through how to rewrite algebraic expressions so they look different, but are actually the same thing.


Do you know what a synonym of happy is? There are several. How about joyful, elated or cheerful? You might have listed some others, which is great! Synonyms are an analogy for what we are going to talk about in this lesson, which is rewriting algebraic expressions using structure. Think of the result of this process as being the mathematical synonym of the original expression. Let's see how to do this using some examples.

Rewriting Algebraic Expressions

Example 1

Prompt: Let x + y = 0. Which one of these expressions equals xy?

a) xy

b) x - y = 1

c) x2

d) -y2

Solution: To start this problem, we'll subtract y from both sides of the equation giving us:

x = -y

Now we'll plug this into the equation they want us to find a different version of, which is xy. This results in:


which equals


This is choice ''d'' so we have our answer.

Example 2

Prompt: Suppose x + 4y = x. Which of the following expressions is equal to x - y?

a) xy

b) x - y

c) (xy)2

d) x

Solution: We'll start this problem by subtracting x from both sides of the equation giving us:

4y = 0

This means y is zero! Plugging in 0 for y into the expression they want us to determine the equivalent of, we get:

x - 0 = x

This is choice ''d''.

Example 3

Prompt: We have the expression a + b = b2 - a. Which expression is the equivalent of √(2a) (b)?

a) b2

b) a2

c) 2a

d) √(ab)

Solution: We will continue the trend of solving the given expression for one of the variables. Let's solve for b. Subtracting a from both sides give us:

b = b2 - 2a

Dividing both sides by b results in:

1 = b - 2a/b

Subtracting b from both sides is next.

-b = -2a/b

Multiplying both sides by b, we get:

-b2 = -2a

Finally, we reach what b equals.

b = √(2a)

Plugging in this expression for b into the second equation we were given results in:

√(2a)⋅ √(2a)

This simplifies to 2a, which is choice ''c''.

Example 4

Prompt: We have the expression x - 2y = xy. Which of the following choices is the equivalent of (x + y) / x?

a) (x + y) / (x - y)

b) x2y

c) (x - y) / (x + y)

d) (x + 3) / (x + 2)

Solution: Do you know the pattern we have been following? If so, you might already know that we are going to solve the first expression for y.

Let's add 2y to both sides, which results in:

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