In the previous section we explored how to take a basic algebraic problem and turn it into a proof, using the common algebraic properties you know as the "reasons" in the proof. In this section, I'll show you a couple examples that use those properties, plus the concept of substitution. Technically, "substitution" is considered to be a method rather than a property, but most textbooks will refer to the "substitution property," and we will do the same here. This idea is very similar to the "Transitive Property," which we will look at in a later section.

The substitution property says that if x = y, then in any true equation involving y, you can replace y with x, and you will still have a true equation. How can we use that in a proof? Here's an example:

Prove: if x + y = 3 and y = 13, then x = -10.

Since this is a proof problem, we're going to set up a two column format with Statements and Reasons. In this problem, how many pieces of information were given to us? Two, right? We were told that x + y = 3, and we were told that y = 13. Great! That makes the first two lines of our proof easy!

STATEMENT                          REASON
1. x + y = 3                       Given
2. y = 13                          Given

Hopefully at this point, you know what to do next; we can substitute 13 in place of y in the first equation. And the reason that we can do this is substitution. So we'll do this:

STATEMENT                          REASON
1. x + y = 3                       Given
2. y = 13                          Given
3. x + 13 = 3                      Substitution property 

Now so far in doing these algebraic proofs, every step has depended on the previous step. But in this case, our step number 3 depended on both steps 2 and 1, right? We used the Substitution property to combine those two equations into something new. Therefore, we can't just state "Substitution Property" - we also have to specify that we were using two previous steps:

STATEMENT                          REASON
1. x + y = 3                       Given
2. y = 13                          Given
3. x + 13 = 3                      Substitution property (1,2)

At this point, we've already simplified this to something very straightforward, so we'll finish the proof now.

STATEMENT                          REASON
1. x + y = 3                       Given
2. y = 13                          Given
3. x + 13 = 3                      Substitution property (1,2)
4. x + 13 - 13 = 3 - 13            Subtraction property of equality
5. ∴ x = -10                       Combining like terms

Let's try another! Here's a problem.

Prove: If x + y = 10, and x + 2y = 20, then x = 0.

Here is a proof, in its entirety. Can you follow the reasoning? Note that this is not the only way to do the proof; there are multiple possibilities, and this is certainly not the shortest way to do it, so you might want to try different ways to see if you can find a process that uses less than 12 steps!

STATEMENT                          REASON
1. x + y = 10                      Given
2. x + 2y = 20                     Given
3. x + y - x = 10 - x              Additive property of equality (1)
4. y = 10 - x                      Combining like terms
5. x + 2(10 - x) = 20              Substitution property  (2,4)
7. x + 20 - 2x = 20                Distributive property
8. -x + 20 = 20                    Combining like terms
9. -x + 20 - 20 = 20 - 20          Subtraction property of equality
10. -x = 0                         Combining like terms
11. -x(-1) = 0(-1)                 Multiplicative property of equality
12. ∴ x = 0                        Arithmetic