Algebra

Manipulating Equations I

Equations are really useful, and you need to know how to manipulate them. Consider the following equation:

2 = 1 + 1

There's no harm in adding 1 to both sides of the equation, as it will still be valid:

2 + 1 = 1 + 1 + 1

2 + 1 = 3

To prove that this is possible we can make the arbitrary form of an equation:

x = x

If we add 1 to both sides we get:

x + 1 = x + 1

And we know that this is valid because both expressions (x + 1 and x + 1) is the same.

We could also add 2:

x + 2 = x + 2

Or 3:

x + 3 = x + 3

Or any number. No matter what you add to both sides of the equation, the equation would still be valid. So the rule is, is that if you have an equation:

x = x

You can add any number to both sides of it (let's say c)

x + c = x + c

We could also give negative y the following property:

- y = c

Where c is the negative form of y. We can substitute - y where there is a c to get:

x + c = x + c

x + (-y) = x + (-y)

x - y = x - y

So that means we can also subtract any number from both sides of an equation. Let's apply our knowledge on a simple equation:

Question 4: 2 - x = x + 2

We could add both sides of the equation by x to get:

2 - x + x = x + 2 + x

We know that -x + x is 0, so we get:

2 = x + 2 + x

We can also subtract both sides by 2 to get:

2 - 2 = x + 2 + x - 2

2 - 2 = x + x + 2 - 2

0 = x + x

And so now we know that the only possible value of x is 0. This makes sense because if we substitute x = 0 into our original equation, we get:

2 -x = x + 2

2 - 0 = 0 + 2

2 = 2