Welcome to an introduction to proof
using properties of equality. The goal to solve an equation to
illustrate a two column proof. Normally when we solve equations we don’t justify each step of the process. However, we could. Justifying each step is an important part of a formal proof. So to introduce the idea of a two column proof in Geometry, we will solve an equation and justify each step with the appropriate property of equality. Let’s go ahead and start by reviewing the properties of equality. We won’t go through all of these but you may want to pause the video to take a look at each of these
properties just in case you’re not familiar with the names of these, even though you’ve used these over and over again to solve different types of equations. Let’s take a look at our two column proof. Here we want to solve the given equation and justify each step in a two column format. So on a two
column proof, we normally have our steps, or procedures on the left and then our reasons, or justifications on the right. And to keep things organized, it’s often helpful to number each step and each region. So number one would be the given equation, so over here on the right we’ll just say given. Now if we want to solve this equation, the next step would be to distribute the five. So let’s go ahead and do that and then we’ll write the justification, or reason on the right. So if we distribute would have ten-x, minus twenty, plus thirty-six equals six-x plus forty. And our justification would be the distributive property of equality. And now I’m just going to write distributive property The next step would probably be to
combine these like term here, so let’s go ahead and do that. That’ll be step three. So we’ll have ten-x. This would be plus sixteen, equals six-x plus forty. And the justification, or reason would be combining like terms. The next step we probably want to get x on one side of the equation. And by habit it’s nice to have the variable on the left side of the equation. So in order to eliminate this positive six-x, we’d have to subtract six-x on both sides. Well if we subtract six-x from ten-x we would have four-x. And then plus sixteen equals forty. So what we did here is we subtracted six-x on both sides of the equation. So our justification is the subtraction property of equality. Now to isolate this x, the next step would be to subtract sixteen on both sides, that would give us four-x equals twenty-four. If the justification is the same reason as number four, the subtraction property of equality. And then for step six, I’m going to go and show a little bit of work here. We’re going to divide both sides by four, and the justification is the division property of equality. And our last step is just to simplify, so we would have x equals six. So our justification would be just simplification, or simplify. Now the new part about this was not solving the equation, the new part is developing the idea of a two column proof, which includes justification for each
step. And this is how we’ll go about setting up geometric proofs throughout the remainder of the course. So in next video, we’ll take a look at proofs using geometric topics. I hope you found this intro helpful. Thank you for watching.