04 Equivalent Expressions

Equivalent expressions

We have seen that the way of getting simpler expressions from a given expression is done through symbolic transformations. We follow the PEMDAS rule to make sure we do the operations present in the expressions in the right order.

But why use these transformations? One is obviously to simplify expressions, but the other is to understand and get equivalent expressions. Let’s understand what they mean. 3 x 4 can be written as 4 + 4 + 4 because we can transform 4 + 4 + 4 into 3 x 4 using symbolic transformation. Since we can simplify them, we see that they are both equal to 12. This is helpful to know since unlike numerical expression, we do not really know the value of algebraic expression due to the presence of variables. So, when we use symbolic transformation on algebraic expressions, we always get an expression that has the same value as the original. For example, we can change π‘₯ + π‘₯ + π‘₯ to 3π‘₯ using the distributive property (π‘₯(1 + 1 + 1) = 3π‘₯). No matter what value we put in for π‘₯, both π‘₯ + π‘₯ + π‘₯ and 3π‘₯ will give us the same answer. Suppose π‘₯ is taken to be 25, then 25 + 25 + 25 is 75, and 3 x 25 is also 75. If π‘₯ is 3 then 3 + 3 + 3 is 9 and 3 x 3 is also 9. Such expressions are called equivalent expressions.

There are a few properties that can be used for symbolic transformations. They are mentioned below.

Distributive Property

We have already mentioned this briefly before, but let’s go through it in more detail. The distributive property is the property by which expressions under multiplication/division can be written in the form of addition/subtraction. Generalizing, it simply means: a(b + c) = ab + ac
(π‘₯ + z)/y = (π‘₯/y) + (z/y)

Here is an example. The area of a purple rectangle is 4 x a because the dimensions of the rectangle are 4 units in width and β€œa” units in length. The area of a red rectangle is 4 x 3 or 12 because the red rectangle is 4 units wide and 3 units long. 1.9 The total area of the figure is 4a +12, which we get by adding the area of both the rectangles.

Looking at it in another perspective, if we look at the dimensions of the whole rectangle (red + purple), we find that it is a + 3 units long and 4 units wide. Thus the area of the whole rectangle is 4 x (a + 3). Both 4 x (a+3) and 4a +12 must be equal since we are talking about the area of the same rectangular image. So they are equivalent expressions. If we want to know the real value of the area of the whole rectangle, then the value of β€˜a’ must be substituted. Let’s suppose that the value of a is 5 units. Then the value of the expression is 4 x (5 + 3) or 32. The value is the same even if we put a = 5 in the expression 4a + 12.

We can also go from one expression to another without having to draw images. In 4a +12, 4 is a common multiple of both 4a and 12, so we can take 4 common from the two to get 4(a + 3). In 4(a + 3), we need to multiply both a and 3 with 4, which gives us 4a + 12.

Easy, right? Practice a bunch of such questions and you will be an expert in it!

Commutative Property

The commutative property is another property that helps us find equivalent expressions. Basically, it means that we can switch the order of two things if we’re adding or multiplying them. For example, 2 + 3 is the same as 3 + 2. Easy, right? But this doesn’t work for everything. If we’re dividing, we can’t switch the order. 10/5 is not the same as 5/10.

Essentially, the commutative property is valid in operations such as addition and multiplication whereas is invalid for operations such as subtraction and division. Let’s test the commutative property with numerical values. 2 + 8 = 8 + 2
2 - 8 β‰  8 - 2
Seems pretty obvious right? We see that subtraction is not commutative since -6 and 6 from above are clearly different. But we can represent subtraction as the addition of the negative quantity. Now that this is in the form of addition, we can apply the commutative property and see that we get the same answer. 2 - 8 = 2 + (-8) = (-8) + 2 = -8 + 2
So, commutative property results in the formation of equivalent expressions. Some examples are:

5a + 7 = 7 + 5a
3b - 8 = -8 + 3b (3b - 8 β‰  8 - 3b)

Are they always equivalent?

In case of equiva;ent expression, we get the same value for all different values of variables we replace in the expressions. For example, in the case of 4 x (a+3) and 4a +12, we get the same value for both when a =1, 2, 3, 5, 10, …..and all other values. But there are cases in which two expressions give the same value but only for one particular value for a variable. For example, 5π‘₯ and π‘₯ - 8 both amount to -10 when we replace π‘₯ with -2:
5π‘₯ = 5 x -2 = -10
π‘₯ - 8 = -2 - 8 = -10
If you replaced the value of π‘₯ with any other number besides -2, you would not get the same value for the two expressions. Let’s see the value of the two expressions when π‘₯ = 4.

5π‘₯ = 5 x 4 = 20
π‘₯ - 8 = 4 - 8 = -4

So, these expressions are only equal at π‘₯ = -2. Expressions are considered equal when they amount to one value at one particular point, like in the example above. On the other hand, equivalent expressions will amount to the same number for all values of variables. We can check if the expressions above are equal at another point as well.