Operations That Produce Equivalent Equations

Last Updated on February 14, 2024.

We can see quite a few operations that produce equivalent equations. In this part, we’ll look at these two: addition & subtraction.

Here we talk about Adding identical quantities to both sides of equations.

This lesson is provided by Onsego GED Prep.

Video Transcription

If we add the same quantities to both the left and the right sides of a given equation, the solution set will not be changed. Let’s see:

If a = b, and we’ll add c to both sides of this equation, we’ll have an equivalent equation.
a + c = b + c.

Let’s take a look if this really works.

Consider the equation x −4 = 3

We can see that the number 7 (seven) is the only solution here in our equation.

So, let’s, for example, add 4 (four) to both sides of our equation and see if the equation that’s the result is equivalent to the original x−4 = 3.

x −4 = 3 is our given equation.
x − 4 + 4 = 3 + 4 (we’ve added 4 to both sides of our equation.
x = 7 ( just simplify both sides of our equation).

Well, 7 is the solution (and the only one) of our equation x = 7. Therefore: the equation x = 7 is the equivalency of the original equation x − 4 = 3 (they both have identical solutions).

This is an important point. If we add the same amount to both the left and the right sides of an equation, its solution won’t change.

It is a fact as well that if we subtract the same quantity from both equation sides, an equivalent equation will be the result.

Subtracting the same quantities from both equation sides.

Again, if we subtract the same quantities from both equation sides, our solution set won’t change.

So let’s see:

If a = b, and we subtract c from both sides of our equation, we’ll have the equivalent equation:
a − c = b − c.

Let’s see if this also works as stated:

Consider this equation: x + 4 = 9.

By inspection, the number 5 is this equation’s only solution.

Now, let’s subtract 4 (four) from both sides of our equation and check if the resulting equation still is equivalent to our original x+ 4 = 9.

x + 4 = 9 is our given equation.
x + 4 − 4 = 9 − 4 (we subtract 4 (four) from both sides of our equation.
x = 5 (just simplify both sides of our equation.

The number 5 is actually the only solution to our equation, so x = 5. Therefore, the equation x = 5 is the equivalency of our initial equation x + 4 = 9 (both equations have the same solution).

The important point is that if we subtract the same amount from both the left side and the right side of an equation, its solutions will not change.