Learn about systems of equations Video

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Learn about systems of equations

Welcome to the presentation on Systems of Linear Equations. So let’s get started and see what this is all about. So let’s say I had two equations now. The first equation, let me write it as 9x-4y=(-78), and the second equation I will write as 4x+y=(-18). Now what we’re going to do now is we’re actually going to use both equations to solve for x and y. We already know that if you have one equation that has one variable, it’s very easy to solve for that one variable. But now we have two equations, you count these two constraints and we’re going to solve for both variables. And you might be looking to it, well how does that work? Is it just magic that two equations can solve for two variables? Was not because you can actually rearrange each of these equations so that they look kind of a normal y=mx+b format. I’m not going to draw these actual two equations because I don’t know what they look like. But if this was a coordinate axis and I don’t know what that first line does look like, we could do another module where we’ll figure out. But let’s just say for sake of argument that, that first line all the x’s and y’s that satisfy 9x-4y=(-78). Let’s say it looks something like that, and let’s say all of the x’s and y’s that satisfy that second equation; 4x+y=(-18). Let’s say that look something like this. So on a line is all of the x’s and y’s that satisfy this equation and on the green line are all the x’s and y’s that satisfy this equation. But there’s only one pair of x‘s and y’s that satisfy both equations and you can guess what that is. That’s right here, right? Whatever that point is, notice it’s on both lines. So whatever x and y that is would be the solution to this system of equation. So let’s actually figure out how to do that. What we want to do is eliminate a variable because if we’re going to eliminate a variable then we can just solve for the one that’s left over. And the way to do that, let’s see, I feel like eliminating this y. I think we got an intuition for how we can do that later on. And the way I’m going to do that is I’m going to make it so that when I add this to this, they cancel out. Well they don’t cancel out right now so I have to multiply this bottom equation by four and I think it will be obvious why I’m doing it. So let’ multiply this bottom equation by four and I get 16x+4y=, let’s see; 40+32-72. All I did is I multiplied both sides of these equations by four. And you have to multiply every term because it’s distributed properly on both sides for the view to one side to the other. Let me rewrite that top equation again; 9x-4y=(-78). Well now if we we’re to add these two equations, when you add equations just add the left side and you add the right side. But when you add, you have 16x+9x=25x, 16+9; 4y-4 that just equal 0, so that’s plus 0=, and then we have (-72), (-78). So let’s see, that’s minus, 140, 150; (-150), alright just adding them all together. So we have 25x=150, well we could just divide both sides with 25 or multiply both sides by one over 25, it’s the same thing. You get x equals, that’s a (-150); x=(-6), there we solved the x-coordinate. Now to solve the y-coordinate we can just use either one of these equations of the top. So let’s use this one since it’s a little bit large and generally simpler. So we just substitute the x back in there and we get 4x(-6)+y=(-18). Let’s go up here, 4x(-6) we get (-24)+y=(-18). And then get y=24-18, so y=6. So these two lines or these two equations, you can even say, intersect at the point axis (-6) and y is +6. So they actually intersect some place around here to that I draw the line probably look something like that. That’s pretty, now we actually solve for two variables using two equations. Let’s see how much time I have, I think we have to do another problem. So let’s say I had the point, (-7x)-4y=9 and the second equation is going to be x+2y=3. Now if I were just doing this just as fast as possible, I’ll probably multiply this equation times seven and it will automatically cancel out. But that’s easy way, I’m going to show you that sometimes you might have to multiply both equations, actually not in this case. So let’s just do it the fast way, real fast. Let’s multiply this bottom equation by seven. And the whole reason I want to multiply it by seven because I want this to cancel out with this. If you multiply it with seven, you get 7x+14y=21. Let’s write that first equation down, again (-7x)-4y=9. Now just add this is a positive to 7x just looks like a negative though. Okay so that’s zero, 14-4y+10y=30; y=3. And not we just substitute back either one of these equations, let’s do that one; x+2 times, 2 times y, that’s 2 times 3; x+6=3. So we get x with a (-3). That one was super easy, the intercept, hope I didn’t do it so fast. Well you can pause and you watch again if you have to. So these two lines intersect at the point (-3,3). Let’s do one more. This one’s harder, I think it will. So it’s (-3x)-9y=66, we have (-7x)+4y=(-71). So here it’s not obvious, what we’ll have to do is if we want to cancel out the y’s first. What we do id we try to make both of them equal to the least common multiple of nine and four. So if we multiply the top equation by four we get (-12x)-36y=, four times six, 240 plus 24 is 264. And we multiply the second equation by nine. So it’s (-63x) plus 36y is equal to, let’s see. That’s 639, big numbers. Now you add the two equations, (-12) minus 63, that’s (-75x), this cancel out, equals 264. Let’s see, what’s 639 minus 264. See I do this in real time; I don’t use some kind of solution thing, solution manual or something. Let’s see 13 and five, 70, I don’t know if I’m right but we’ll see. Since it’s actually the negative on 639, this is (-375) and I know that 75 goes in to 300 four times so x is equal to five because 75 times 5 is 375, we just divided both sides by 75. So if x is 5, we just substitute back in to let’s use this equation. So we get (-3) times 5 minus 9y is equal to 66; so we get (-15) minus 9y is equal 66, (-9y) is equal to 81 and then we get y is equal to (-9). So the answer is (5,-9). I think you’re ready to do some systems of equations now. Have fun!

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Welcome to the presentation on Systems of Linear Equations. So let’s get started and see what this is all about. So let’s say I had two equations now. The first equation, let me write it as 9x-4y=(-78), and the second equation I will write as 4x+y=(-18). Now what we’re going to do now is we’re actually going to use both equations to solve for x and y. We already know that if you have one equation that has one variable, it’s very easy to solve for that one variable. But now we have two equations, you count these two constraints and we’re going to solve for both variables. And you might be looking to it, well how does that work? Is it just magic that two equations can solve for two variables? Was not because you can actually rearrange each of these equations so that they look kind of a normal y=mx+b format.

I’m not going to draw these actual two equations because I don’t know what they look like. But if this was a coordinate axis and I don’t know what that first line does look like, we could do another module where we’ll figure out. But let’s just say for sake of argument that, that first line all the x’s and y’s that satisfy 9x-4y=(-78). Let’s say it looks something like that, and let’s say all of the x’s and y’s that satisfy that second equation; 4x+y=(-18). Let’s say that look something like this. So on a line is all of the x’s and y’s that satisfy this equation and on the green line are all the x’s and y’s that satisfy this equation. But there’s only one pair of x‘s and y’s that satisfy both equations and you can guess what that is. That’s right here, right? Whatever that point is, notice it’s on both lines. So whatever x and y that is would be the solution to this system of equation. So let’s actually figure out how to do that.

What we want to do is eliminate a variable because if we’re going to eliminate a variable then we can just solve for the one that’s left over. And the way to do that, let’s see, I feel like eliminating this y. I think we got an intuition for how we can do that later on. And the way I’m going to do that is I’m going to make it so that when I add this to this, they cancel out. Well they don’t cancel out right now so I have to multiply this bottom equation by four and I think it will be obvious why I’m doing it. So let’ multiply this bottom equation by four and I get 16x+4y=, let’s see; 40+32-72. All I did is I multiplied both sides of these equations by four. And you have to multiply every term because it’s distributed properly on both sides for the view to one side to the other.

Let me rewrite that top equation again; 9x-4y=(-78). Well now if we we’re to add these two equations, when you add equations just add the left side and you add the right side. But when you add, you have 16x+9x=25x, 16+9; 4y-4 that just equal 0, so that’s plus 0=, and then we have (-72), (-78). So let’s see, that’s minus, 140, 150; (-150), alright just adding them all together. So we have 25x=150, well we could just divide both sides with 25 or multiply both sides by one over 25, it’s the same thing. You get x equals, that’s a (-150); x=(-6), there we solved the x-coordinate.

Now to solve the y-coordinate we can just use either one of these equations of the top. So let’s use this one since it’s a little bit large and generally simpler. So we just substitute the x back in there and we get 4x(-6)+y=(-18). Let’s go up here, 4x(-6) we get (-24)+y=(-18). And then get y=24-18, so y=6. So these two lines or these two equations, you can even say, intersect at the point axis (-6) and y is +6. So they actually intersect some place around here to that I draw the line probably look something like that. That’s pretty, now we actually solve for two variables using two equations. Let’s see how much time I have, I think we have to do another problem.

So let’s say I had the point, (-7x)-4y=9 and the second equation is going to be x+2y=3. Now if I were just doing this just as fast as possible, I’ll probably multiply this equation times seven and it will automatically cancel out. But that’s easy way, I’m going to show you that sometimes you might have to multiply both equations, actually not in this case. So let’s just do it the fast way, real fast. Let’s multiply this bottom equation by seven. And the whole reason I want to multiply it by seven because I want this to cancel out with this. If you multiply it with seven, you get 7x+14y=21. Let’s write that first equation down, again (-7x)-4y=9. Now just add this is a positive to 7x just looks like a negative though. Okay so that’s zero, 14-4y+10y=30; y=3. And not we just substitute back either one of these equations, let’s do that one; x+2 times, 2 times y, that’s 2 times 3; x+6=3. So we get x with a (-3).

That one was super easy, the intercept, hope I didn’t do it so fast. Well you can pause and you watch again if you have to. So these two lines intersect at the point (-3,3). Let’s do one more. This one’s harder, I think it will. So it’s (-3x)-9y=66, we have (-7x)+4y=(-71). So here it’s not obvious, what we’ll have to do is if we want to cancel out the y’s first. What we do id we try to make both of them equal to the least common multiple of nine and four. So if we multiply the top equation by four we get (-12x)-36y=, four times six, 240 plus 24 is 264. And we multiply the second equation by nine. So it’s (-63x) plus 36y is equal to, let’s see. That’s 639, big numbers. Now you add the two equations, (-12) minus 63, that’s (-75x), this cancel out, equals 264. Let’s see, what’s 639 minus 264. See I do this in real time; I don’t use some kind of solution thing, solution manual or something.

Let’s see 13 and five, 70, I don’t know if I’m right but we’ll see. Since it’s actually the negative on 639, this is (-375) and I know that 75 goes in to 300 four times so x is equal to five because 75 times 5 is 375, we just divided both sides by 75. So if x is 5, we just substitute back in to let’s use this equation. So we get (-3) times 5 minus 9y is equal to 66; so we get (-15) minus 9y is equal 66, (-9y) is equal to 81 and then we get y is equal to (-9). So the answer is (5,-9). I think you’re ready to do some systems of equations now. Have fun!

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