Equivalent Expressions

 


Equivalent Expressions: Definition

Equivalent expressions are two or more expressions that may be made of different terms and numbers but produce the same results regardless of what the value of the variable is. In other words, for every number you sub into the variable spot (can be x or a or h) , both expressions give the same value.

Expand and simplify the expressions and then show that the original version and the simplified versions are equivalent for two variables of x (you choose any values of x).

The solutions in the above animation are repeated with a more detailed explanation below.

Question 1.1

4(3x + 2) - 5

Firstly, look at the expression and see what we can expand. We expand 4(3x + 2) on its own using our methods from previous lessons.

(Remember the Distributive Property, a(b+c) = ab +bc )

4(3x) + 4(2) =

(4)(3)x + 4(2) =

12x + 8, now we put this back in to our original expression

4(3x + 2) - 5 =

[ 4(3x + 2) = 12x + 8 ]

12x + 8 - 5 =

12x + 3

Now we have the original expression 4(3x + 2) - 5 and the simplified expression 12x + 3 and we have to show they are equivalent.

Remember, Equivalent definition : Two or more expressions that have equal results for the same value of the variable (variable in this case is x).

All we have to do is pick ANY two numbers for x. Let's go for 1 and 3.

First, we lets sub x = 1 into both expressions,

4(3x + 2) - 5

4(3(1) + 2) - 5 =

We will gradually start to combine steps of order of operations and the three definitions of properties for speed's sake.

4(5) - 5 =

20 - 5 = 15

Now the other expression,

12x + 3 =

12(1) + 3 = 15

So they are the same for x = 1. Let's check x = 3.

4(3x + 2) - 5

4(3(3) + 2) - 5 =

4(9 + 2) - 5 = 39

12x + 3

12(3) + 3 =

36 + 3 = 39

So, for both expressions we had the same results for the two different values of x. The work we did to simplify the original expression was proof in itself that the original and simplified versions were equivalent. Subbing in different values of x just checks that we were right.

When simplifying an expression, you can check that you simplified it correctly by subbing a value into the variable for the original expression from a question and simplified version that you worked out!

As a “fun” exercise, try subbing in a massive or obscure number into the two expressions we just worked with (you can use a calculator for this one) and see what happens! They will still be equivalent! Try something like x = 3,456 or x = (147/16)

Question 1.2

opening bracket
5(3x + 3) - 3
3
closing bracket
  - 7

This expression can look quite daunting, but as always, all we have to do is remember our rules and methods. The order of operations, and the three properties (associative, commutative and distributive).

It might help to think of this as “something being divided by 3 then minus 7”,

As order of operations tells us (BEDMAS), deal with the division first,

let’s split this up from the original expression,

5(3x + 3) - 3
3

Now, lets deal with the numerator first.

5(3x + 3) - 3

Expand out our brackets

5(3x) + 5(3) - 3 =

15x + 15 - 3 =

15x + 12 is our numerator, so we have,

15x + 12
3

Distributive Property,

15x
3
 + 
12
3

= 5x + 4,

Important to remember to sub this all back into the original expression, so

opening bracket
5(3x + 3) - 3
3
closing bracket
  = 5x + 4
opening bracket
5(3x + 3) - 3
3
closing bracket
  - 7 =

= 5x + 4 - 7 = 5x - 3,

Now we have our simplified version, let's check they are equivalent by subbing in two value of x. Let’s try 2 and 4.

Working through this we will start to assume there is an understanding about some of the steps of the order of operations and the three properties. However, if you're unsure, never hesitate to work through every step yourself.

x = 2

opening bracket
5(3x + 3) - 3
3
closing bracket
  - 7 =
opening bracket
5(3(2) + 3) - 3
3
closing bracket
  - 7 =
opening bracket
5(9) - 3
3
closing bracket
  - 7 =
opening bracket
45 - 3
3
closing bracket
  - 7 =
opening bracket
42
3
closing bracket
  - 7 =

14 - 7 = 7

Now check the simplified version.

5x - 3

5(2) - 3 =

10 - 3 = 7

So, we are right for x = 2, let's check x = 4

x = 4

opening bracket
5(3x + 3) - 3
3
closing bracket
  - 7 =
opening bracket
5(3(4) + 3) - 3
3
closing bracket
  - 7 =
opening bracket
5(12 + 3) - 3
3
closing bracket
  - 7 =
opening bracket
5(15) - 3
3
closing bracket
  - 7 =
opening bracket
75- 3
3
closing bracket
  - 7 =
opening bracket
72
3
closing bracket
  - 7 =

24 - 7 = 17

Now check the simplified version.

5x - 3

5(4) - 3 =

20 - 3 = 17

Bingo! 17 = 17