# Algebraic Proofs | It All Adds Up

Hi there! Sorry for the delay on this post… High School has kept me very busy.

For the rest of the year, most of our Geometry units will center around writing proofs. A proof is a logical way of proving that a statement is true, and it consists of statements and reasons that logically follow each other. Today, I’m going to teach you about writing algebraic proofs, which involve solving for a variable. Most of the year, though, we will focus on writing geometric proofs, which involve lines, angles, etc. With this post, you’ll get a general idea for what a proof is. However, geometric proofs are something that really have to be taught in a class, or be taught over the course of several days. I can’t really cover that here, and that puts It All Adds Up at a dilemma.

Here’s the plan: in order for you to understand future editions, you will need to research geometric proofs on your own. Learn what they are, how they work, and how you form a proof (since there are so many possibilities for statements and reasons). For It All Adds Up, I’ll be going over the aspects that can be reasons or statements, along with some tips for writing proofs with them.

For today, though, we’ll start simple. We already know how to solve for a variable, such as in x + 68 = 20. Now, each step that we would use to solve that becomes a statement.

Take the 4 step proof above. Each step, from top to bottom, shows how we solve for the value of x. We know that x does equal 38, but how would we prove it? For algebraic proofs, there are several properties that you’ll need to know.

Properties of Equality (For Algebraic Proofs)

Addition POE: When the same value is added to both sides of an equation

Subtraction POE: When the same value is subtracted from both sides of an equation

Division POE: When both sides of an equation are divided by the same value

Multiplication POE: When both sides of an equation are multiplied by the same value

Other Properties (For Algebraic Proofs)

Distributive Property: When a number outside of a parenthesis is distributed to each value within the parenthesis

Now, there are many other properties, postulates, and theorems to come, but for this week, I’ll be focusing on simply solving for a variable (for the purpose of getting you used to proofs) and thus, the properties stated above will be sufficient. I’ll be presenting you with numerous others in future editions.

So let’s fill in the proof.

Here’s another aspect I want to focus on: the given. The given is the starting point of any logical process, and will always be the first step in the proof. It’s what you are, well, given. For example, we’d be given the first equation and then asked to solve for x. As for the other properties, you always want to ask yourself: How did I get from this step to the next step? Your answer will vary- it might be adding 72 to both sides, it might be dividing both sides by three.

Usually, when asked to write a proof, you’ll also be provided with a solution. (This isn’t the case with algebraic proofs, though). That will be your last step. For algebraic proofs, your last statement will be ____ = ____, but with geometric proofs, it can vary. You might have to prove two angles are supplementary, prove two segments are congruent, or two lines are parallel.

I’m learning right along with you- and believe me, it takes me a while to process everything too. But overall, this is a proof. And everything you’ll learn this year from here on out will build on this format. Once you know what a geometric proof can be, I’ll supply you with possible statements and reasons, as well as specific rules and patterns that apply. Do remember though: a proof can have many different solutions.

Thanks so much for reading. Have a great long weekend!

Gabe