Algebra Know-It-ALL

174 Equations and Inequalities

Now suppose we have some variable or quantity. We don’t know its numerical value, but we

can find it if we want. Let’s call that quantity x. We can add x to both sides of any equation

we might come across. For example,

5/2+x= 20/8 +x

We can subtract the same quantity from both sides, or from all parts, of an equation and get

another valid equation. Here’s an example:

5/2= 10/4 = 20/8

If we subtract 1/2 from each part of this, we get

5/2− 1/2 = 10/4 − 1/2 = 20/8 − 1/2

Simplifying this gives us the three-way equation 2 = 2 = 2.

Adding or subtracting equations

Just as we can add a quantity to, or subtract it from, all parts of an equation and get another

valid equation, we can add or subtract entire equations to or from each other. Here’s an

example with numbers. Suppose we see these two equations:

5/2= 20/8

and

3/2= 12/8

If we add these two equations to each other by adding their left and right sides individually,

we get

5/2+ 3/2 = 20/8 + 12/8

which simplifies to 8/2 = 32/8, and further to 4 = 4. We can subtract the second equation

from the first, and get

5/2− 3/2 = 20/8 − 12/8

which simplifies to 2/2 = 8/8, and further to 1 = 1.

Multiplying through by a quantity

We can multiply each side, or all parts, of an equation by the same quantity and get another valid

equation. This is called multiplying through. Let’s look at our original numeric example first:

5/2= 20/8