Learn about Properties of Equality In this lesson, let's learn about properties of equality when it comes to algebraic and geometric equations. So, we’ll do two problems. The first one wants us to solve this equation and write a justification for each step. Before we go there, let's discover quickly what are the different properties of equality. We may know how to do these anyway but it's nice to know where they come from. So for example, the addition property says, if we have two things that are equal, if A = B, then if I add C to both sides, I'm allowed to do that and the equality would still be correct. Similarly, I can subtract the same number from both sides of the equation. I can multiply in both sides. I can divide on both sides. When I divide, I cannot divide by zero of course. Then, if A = B and B = C, then A = C. If 1 = 2, 2 = 1, then 1 = 1. In substitution property which means if two things are equal, I can replace one with the other. So these are the things that are called properties equality and we’ll use these to solve the equations. So for example, we’ve got (-6 = 2N + 2). The justification is this is the given statement. So now let's try and solve this. So (-6 = 2N + 2), what we’re going to do is subtract two from both sides, (+ 2N + 2 – 2). What I'm doing is subtracting two from both sides; that, I'm allowed to do because of this subtraction property. Let me create some more space for us. So now that I have this, let's simplify. (-6, -2 is -8) this is 2N + 2 -2 get cancelled. So now, let's divide both sides by two. This is the division property, (-8/2 is -4 = 2N/2 is N). So, to this again, we simplify and I have the answer and equals (-4). Notice all we did was basically take the equation and use the different properties to get to the answer. Let's do one more problem. It says Meredith’s breakfast cereal has 102 calories per serving. The equation C = 9f + 90 relates the grams of fat (f) in one serving to the calories C in the serving. How many grams of fat in one serving of the cereal? We’re given the number of calories. Solve the equation for f and justify each step. So let's solve this equation for f. How would we do this? This is the given statement. So the first thing I'm doing is subtracting 90 from both sides. That is the subtraction property. So what am I left with? C – 90 = 9f. This is simplified. Then what do I do? I divide both sides by f which is the division property and I am left with C – 90/f =—sorry—divided by nine. We’re dividing by nine. We are left with C – 90/9 = nine and nine get cancelled f. So, that was by a division property and then I simplified. So what is the answer that I'm left with? F = C – 90/9, that’s my solution for f. Now in order to get how many grams of fat are in one serving, let's replace f with 102 calories. So, f remains f equals C becomes 102 calories minus 90/9 which is equal to 102 – 90 is about 12/9 which is nine, one, 30, 27, 1.33 grams. How many grams of fat? 1.33 grams, f = 1.33 grams. The way did that is we substituted C for 102. That’s substitution property. If we know C = 102, instead of C, I wrote 102 and then solve this to get 1.33 grams of fat.