The Nature of Equations
An equation is like a balanced scale in that each side has to be equal. Consider the following:
3 + 2 = 5
In this rather simple equation, we see that each side will simplify to the same thing no matter what. If we change something on one side, we have to do the same on the other side. For example, what if we add 4 to the left side of the equation above:
3 + 2 + 4 = 5
The sides are no longer equal. If we add 4 to the right side of the equation:
3 + 2 + 4 = 5 + 4
Then the two sides are equal again.
The lesson to be learned here is that if we do something to one side of the equation, it’s fine as long as we do it to the other side as well. This might seem like it’s of little to no importance, but it’s extremely useful when we are trying to solve for the values of variables.
With this in mind, consider the following equation:
a + b = c
If we were so inclined, we could subtract b from each side to have the following:
a + b – b = c – b
a = c – b
Because of our rules for how equations work, we know that a = c – b is true because we started with knowing that a + b = c was true.
We can manipulate equations in similar ways using multiplication and division instead of addition and subtraction. Consider the following:
4(3) = 12
If we wanted, we could multiply both sides by 2, like so:
4(3)(2) = 12(2)
Because we did the same thing to both sides, the equation holds true.
One thing to watch out for when you’re multiplying or dividing like this is that the distributive property applies. Say we have:
2 + 4 = 6
And we want to multiply both sides by 5. We have to enclose the left side in brackets or parentheses since the multiplication by 5 has to apply to the entire left side. For a clearer understanding, look at what happens if we don’t:
2 + 4 * 5 = 6 * 5
2 + 20 = 30
22 = 30
And clearly this is incorrect. But if we use the multiply correctly:
(2 + 4) * 5 = 6 * 5
6 * 5 = 30
Which is clearly correct. The same applies when we divide. If we have this equation:
15 – 10 = 5
And we want to divide both sides by 5, we have to do this:
(15 – 10)/5 = 5/5
5/5 = 5/5
If we don’t, then we get the following:
15 – 10/5 = 5/5
15 – 2 = 1
13 = 1
Which is obviously wrong.
You can also view our Nature of Equations Video for more of an explanation.