 Alright, so now I'm going to talk about solving systems that aren't in like x equal or y equal. If this isn't already done for you, you need to solve for x or solve for y in one equation. Then you take that expression that you just found on this side and plug it into the other equation. That will be important. So now you solve the new equation, so now you've found out what that variable is really equal to. And then you plug that back into one of your original equations, either one, and solve to find out what the second variable is. Look at another system of equations. Remember we're looking for an ordered pair, x equal and y equal. This is our x, this is our y, what are they going to be? So in this case we don't have an x equal or a y equal isolated yet, so we have to do that first step. We have to isolate. Now we have two choices for x's and two choices for y's, whichever equation we want to take. I'm going to take the top equation. Can anybody guess why? It would be because this coefficient on x is just one, so that would be so easy to just solve for x. All I have to do is add across the equal sign. So we started out with x minus 4y equal 4. We're going to add 4y to both sides. So 4 plus that 4y. And now I've got my expression that I can substitute. Remember this is the part that we're going to substitute. We had negative 5x plus 3y equal to 14, but remember we're going to substitute for x. So negative 5 times our x, but this time x is 4 plus 4y. Remember this expression is what we're going to substitute. The rest of our equation was plus 3y equal 14. Now this time, since we have an expression that we are going to multiply by that negative 5, we have to distribute. So negative 5 times 4 would be negative 20, and negative 5 times 4y would be minus 20y. And plus our 3y is equal to 14. And now we're ready to combine like terms. Negative 20 is its own term. Combine those like terms right here, and we have negative 20y plus 3y would be minus 17y is equal to 14. My equal sign over there got messed up. And now we add 20 to both sides so we can take this to the other side. And we have, I'm going to write it over here, negative 17y on that side. Is equal to, you're adding 20 to both sides, we're going to have 34. And dividing by negative 17y is going to be equal to a negative 2. So our ordered pair so far, we know that y is negative 2. That's just the first substitution. Now we have to plug it back in to find the other variable. So we can go back to our original equations. When substitution, when you have to solve one equation for a variable, that kind of makes a nice equation and you haven't really changed the equation. So I'm going to plug mine back into that equation. So I'm going to come in here and say x is equal to 4 plus 4y and substitute in for the y. So x is equal to 4 plus 4 times that negative 2. So we're going to have x is equal to 4 minus 8, negative 2 times 4, then 4 minus 8 would be negative 4. So there's our x and our back up here at our ordered pair. We have negative 4, negative 2. And we don't have to check both equations if we're satisfied with what we did. But let's see if we can see that it really is that for at least one of the equations. So let's plug it into the easy equation. So x which is negative 4 minus 4 times y which is negative 2 if we did it right should be equal to 4. So negative 4 and then negative 4 times that negative 2 would be positive 8 and negative 4 plus 8 would be 4. So we know it satisfies the top equation and if it satisfies the top equation we should feel pretty confident it's going to satisfy the bottom one. You can check it on your own if you want to. We don't have an M equal or P equal equation and we don't have one that M or P has just a coefficient of 1. So we have to work a little bit harder on this one. So we can take either one of the equations and I like small numbers and we haven't tried the second one, the second equation we've always been using the first one. So let's just change things up a little bit. So let's solve this one and 2 is the smallest number so I'm going to solve this one for M. Again why did I do that? Just because these look like nice numbers, 2 is a small number to have to divide by so I thought that would be pretty easy. Actually either one of them works fairly easy in this problem. So if I solve that I have 2M is equal to 20 and then I'm going to add the 10 P to both sides and then I'm going to divide everything by 2. So this will be M, 20 divided by 2 would be 10, 10P divided by 2 would be plus 5P. So I'm going to use 10 plus 5P as what I plug into, to plug my variable into the top equation. So this time I have to go into, I don't have a choice because I solve the second equation, I've got to go into the top equation and I'm going to substitute for M. So 3 times 10 plus 5P, that's what we said M was equal to, minus 15P will equal 30. Again we have to distribute because we have an expression here. So 3 times 10 will be 30, 3 times 5P would be plus 15P, minus 15P is equal to 30. When we combine our like terms they actually cancel each other out because one's a positive P and one's 15P and one's a negative 15P. So this actually tells me that 30 is equal to 30, which is a true statement. It is true that 30 is equal to 30. So what that really tells us that all points, both these lines have all points in common or they are the same line. Alright so let's try a couple points that we know are on either one of these lines and I want to try the point 0, negative 2. So if I plug it into the top equation I have 3 times 0 which is N, minus 15 times negative 2 which is P is equal to 30, but 3 times 0 is just 0 and negative 15 times negative 2 would be positive 30 equal to 30, so it worked for the top equation and just to make sure that you really know this is true let's try it in the bottom equation. So 2 times N which is 0 minus 10 times P which we said is negative 2 should be equal to 20. So 2 times 0 is 0, negative 10 times negative 2 is positive 20 equal to 20, so sure enough it works for both those equations last problem. We have this system right here and 2x minus 2y equals 6, x minus y equals 29. The bottom equation hopefully you can see would be a nice equation to solve for x or for y because they both have coefficients of 1 and let me tell you a little trick this is a positive 1x and a negative 1y, so positive x would be the easiest one to solve for. We don't have to worry about our signs if we divide so x is going to be equal to 29 plus y adding y to both sides. Substituted into the top equation, the other equation we have 2 times our x but remember our x is 29 plus y and then minus 2y equal to 6, so again we have to distribute so 2 times 29 is 58, 2 times y would be plus 2y and then minus 2y equals 6, so combining like terms 2y minus 2y, again they can't reach other out and we have 58 equal to 6 which is a false statement. So if we had a true statement and it was the same line all points are in common then if it's false that would mean that they have no points in common and if you remember talking about special lines, if they have no points in common that means that they never cross or they are parallel lines and I would be fine with no solution.