# Substitution Property

Reference > Mathematics > Introduction to ProofsIn 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

## Questions

^{2}= 6, and y + 3x

^{2}= 20, then y = 2