Hi everyone. Welcome to our lecture on systems of equations. We're going to be focusing on linear equations in two variables. Friendly reminder, what is a linear equation? Linear equation is one of the form Ax plus By equals C. A, B and C, of course, are just some real numbers. We want them not to be zeros that actually have an equation, but this is the standard form of an equation. You can start and solve for y and get different forms. You can get the slope intercept form or point slope form, how you want to manipulate the equation, but realize this is a linear equation and so in particular its graph is a line. We're often interested in solving for these equations. We're often interested to seeing what happens when they turn into inequalities. In this section, we're going to talk about more than one of these equations. We're interested in solving a system, so we're going to define our term. We say a collection of two or more equations is called a system. System is just a fancy way to say I have more than one. For example, as a system of linear equations, I can certainly have 2x plus 3y equals 4 and 3x plus 4y equals 5. Pick your favorite numbers. This is a system of two equations. There's nothing stopping me from listing 100 more, then I would have a system of 102 equations. It doesn't matter. In this particular example, we're going to start small, we're going to keep things linear. You can certainly start adding x^2 plus y^2, whatever. That's a different topic for another day, but we're going to start with two equations in our system and then we're going to be linear. What we're after, what our goal is, we want to find the solution set of the system and this is the points or the ordered pairs x,y that satisfy both equations or all the equations in your system. We're not interested in just solving one or two or three, we want points or x,y coordinates. Let's see a couple of examples and actually solve some of these. Sure, you should know your rules of algebra, but just remember, I'm working here with two lines in these systems of linear equations. In your mind, where do these lines live on the x,y plane and what do these lines do? Well, oftentimes they'll intersect. There's line one and there's line two. Oftentimes they'll intersect and when they do, you have a point that lies on both lines. Any point that lies on both lines will satisfy both equations. What we're really after, our solution set will be the point of intersection, but we will see that not every line has to intersect in just a point. Just as a little hint of what's to follow, can you think of two lines that don't intersect, perhaps on the x,y plane? We'll see examples of this in a second but just keep in mind, we're usually after intersection point or points when they occur. Now, let's solve by graph and graph in such a great tool for these systems of linear equations. I want you to go grab a calculator or open up decimals, and I want you to throw these equations into your calculator. I'll give you a system of two linear equations. Here we go, x plus 2y is 6, 2x minus y is equal to negative 8. These are both lines. Graph them any way you want. In this particular case, I don't want you to graph it by hand. You could and I think the answer here is going to be easy enough, but I rather use a calculator and practice finding your intercept. If they intercept that weird non-traditional point, it's going to be very difficult for you to get that exactly. Go grab decimals and when you do that, pause the video and make sure yours matches. You have to zoom in to the right window of course to find the point of intersection, but decimals is so wonderful that it knows whenever you graph more than one, it'll call out automatically the point of intersection. All you have to do is hover your mouse over the point of intersection and the label, the answer will appear. Our solution set this system is the point negative 2,4. The single point of intersection corresponds to the point where that satisfies both equations. You can test, plug in x equals negative 2 and y is 4 to both equations and it will work out. Let's do another example of a nice system of linear equation. Let's start with 3x minus y is 2 and 2y minus 6x is negative 4. Notice that the x and y's are given a reverse order? That's okay. You can certainly rearrange it if you like. Again, let's do this by graphing. Head over to your calculator or to decimals and graph these two lines and tell me what happened, ready. When you graph these lines, you are going to have a hard time finding that second line and in particular in decimals. You can turn off if you click these colored icons on the left and decimals, you can turn them off and you'll see what you're looking at really is the same line. These equations, while they are different, they have the same solution set. Let's see why that's true. Start with the first one, 3x minus y is 2 and let's multiply both sides by minus 2. Let's send minus 2 on the left and multiply minus 2 on the right. When I do that, you got to distribute and hit both pieces, you get minus 6x plus 2y equals minus 4. Of course a little rearranging will give you that 2y minus 6x is minus 4. They are exactly the same at different equations, but they have the same solution set. The solution set in this particular case is infinitely many points. If I draw a line and draw the same line again, how many times do the two lines meet? All the time. Every single point on that line is a solution to these system of linear equations. The way to hand that back is you can write a set notation, you really just handing back the line 3x minus y is 2. You're heading back infinitely many solutions. Now we've seen that systems can have one solution, that's where the two lines are different and they meet at a single point. They can have infinitely many solutions as this example shows, where they have the same exact line. Let's do one more. As another example of a system of linear equations, let's look at y is 1/2 x plus 2 and x minus 2y is equal to 4. Take a second, graph these two equations, whatever format you need for your particular technology, if you're on a calculator, then often watch yourself a y, go ahead and do it, but if you graph these, what do you get? Let's see. When I graph these in decimals, I get two parallel lines. Now, what do we know about parallel lines? Well, parallel lines have the same slope. Let's see that from the equation. The first equation is in slope intercept form. You can look at the coefficient on the x and tell me that the slope is 1/4. The second equation, however, I have to solve for y to see that so let's go ahead and do that. If I add 4y to both sides, I get x equals 4y plus 4. Let's solve for y moving forward to the other side so we'll subtract 4 and let's switch to the equations, so we get 4y is x minus 4, then divide everything by 4 and we get y is 1/4x plus minus 1. The slope is also one. They have different intercepts. That's why they're different lines. These are not the same lines, but they're parallel. You can see that by looking at their slope. Now, my question to you is, well, if I'm after points of intersection, then I have parallel lines they're never going to touch so they have no points of intersection. That tells me that my solution set is empty. That's the symbol for the empty set, like a zero with a slash through it. There are no solutions. This can happen. This is okay. You have to be comfortable now when you work with more complex objects like a system of equation having no solutions, infinitely many solutions or one solution. In fact, a system of equations that has at least one solution, we call it consistent. System of equations with at least one solution is consistent. We saw this when we had two lines that intersected, this would be consistent. There's one solution and we also saw that if we had lines that were exactly the same, their equations gave the same line. Line one is equal to l2, this is consistent with infinitely many solution. The opposite, the third case that we saw was when we had parallel lines, we had two lines that never touched. In this case, when I have no solutions, we would say we are inconsistent. These words are used to describe the system. I can have a consistent system, which means I have one or infinitely many or I have no solutions in which I'd say the system is inconsistent. A consistent system with exactly one solution, we clarify this and call this independent and a consistent system with infinitely many solutions we would say is dependent. You can have an independent consistent linear system, a lot of vocabulary there for you or a dependent consistent linear system. Graphing is great, however, sometimes the calculator will either not work for whatever reason, it gives us back a decimal, maybe we want close form. We'd like some algebraic way to solve for these linear systems. Graphing equations of systems it does visualize the system tells us how many solutions it has, but we often want a simpler way to do this. We want some algebraic way. The way we're going to talk about now is the substitution method and let's look at an example. Let's start off with a system 3x minus y is 6, and the second equation will be 6x plus 5y is negative 23. Substitution says the following, pick any variable you want and isolate it, solve for it. In this particular case, let's solve for y. I can write y as 3x minus 6. Once I have y, I then substitute it into the second equation or if I had solve for y in the second equation I would substitute it in the first equation, it doesn't matter what you do. You pick one variable and you isolate it and then you plug it in to the second equation. In particular, I get 6x plus 5 parentheses, 3x minus 6 is negative 23. Clean this up a little bit and I get 6x plus 15x minus 30 is negative 20. Now I have a very nice linear equation that I can simplify and solve for x. Keep going, 21x is negative 23 plus 30, that will be positive 7 and solve for x you get 7 divided by 21 that are known as 1/3. Now remember you got to fight your urge to stop here, the x coordinate of the solution is 1/3. However, to find y, I go back to the equation where I isolated y and I plug in and when you do that you get y is 3 times 1/3 minus 6 or 1 minus 6 which is equal to minus 5. Our solution is the coordinate x,y, It's the point of intersection 1/3, negative 5. You can go ahead and check that 1/3, negative 5 does in fact satisfy both of the original equations. Let's do another one. Let's take the equation y is 2x plus 1,000 and let's do 0.05x plus 0.06y is 400. Now, the first equation has done us a little bit of favor, it has already isolated the first variable. We might as well use that and just plug that into the equation for y. When you do that, you get 0.05x plus 0.06 and here we go, you use parentheses 2x plus 1000 equals 400. Now, I'm going to leave this as an exercise, but you can absolutely solve this, distribute as needed. It's okay if you're working with decimals. Now we have another linear system with only x inside of it so we can simplify this. It's okay to work with decimals, get comfortable working with decimals, grab the calculator if you need. We're going to distribute that.6. When we do that, we get.12x plus 60 is still equal to 400. Clean it up a little more, combine like terms, you get.17x is equal to 340 and solve for x, you get x equals 2,000. Remember, I'm not just interested in x, I also want y. How do I get y? Go back to the original equation, plug in x and when you do that you get y is 5,000. Our solution is x equals 2,000, y equals 5,000. Go ahead and check, plug them back into their original equations and you'll see that it in fact works. What does the algebra look like when I have an inconsistent or a dependent system? Watch what happens. Let's start off. Let's look at the linear system 3x minus y is 9 and 2y minus 6x equals 7. Let's go through and solve this using substitution. Doesn't matter which equation you pick. I usually pick the first one just because, but when you isolate y, you get y equals 3x minus 9. I'll then substitute that back into the second equation, I get two parentheses. Don't forget the parentheses, minus 6x is seven. Let's solve for x. We'll distribute the two, you get 6x minus 18 minus 6x equals 7. My goal is to solve for all x's that make this true. The 6x is tend to cancel and all of a sudden I get negative 18 equals 7. Oh boy. Now students get to this point and they panic. They say what happened, I must have done something wrong and they'll go back and do it again and they get the same answer. They get same something like this. The way to interpret this is to ask yourself what values of x make negative 18 equal to 7? When is negative 18 equal to 7? Hopefully, you say wait a minute, it's never equal to 7 and that's okay. Assuming all the other math is right and you get this, just realize that there are no solution. There are no x values that will make negative 18 equal to 7. Don't panic, everything is okay. This is the algebra saying that there's no solutions which graphically what does that mean? Again, you can go graph this if you want. When there's no solutions, there's no points of intersection, when you get some really weird false statement like this, it means the lines are parallel. You got to promise me when you get this, you're not going to panic. You just realize the algebra saying no solution. Let's do one more. Let's look at 1/2x. Let's look at one more. Let's look at 1/2x minus 2/3y equals minus 2 and 4y equals 3x plus 12. Okay, same thing, let's try to solve for the variables. Let's follow the same method as before and isolate y. In this particular case, I think it's easier to work with the second equation. If we divide both sides by 4 , we get y equals 3/4x plus 3. I then substitute that value y into the first equation. I get 1/2x minus 2/3, big parentheses here, 3/4x plus 3 equals minus 2. Distribute the negative 2/3 in. When you do that you get negative 1/2x minus 1/2x minus 2 equals minus 2. The negative 1/2x and the positive 1/2x they cancel and the negative 2 is if I add 2 to both sides, all of a sudden I get 0 equals 0. This is another one of these equations. You could also stop a negative 2 equals negative 2, that's fine too, but this is another one of the things that freak students out and they start to panic. They think something is wrong. Everything is okay. No problem here. Ask yourself the same question to before. What values of x make 0 equal to 0? This statement is an identity. It's always true. Every value of x makes 0 equal to 0. It's always true so the solution set here is all reals. You're in the case when you have infinitely many answers, your solution set is minus infinity to infinity, sometimes written as just R. What this algebra is trying to tell you is that you have the same lines. If you graph this, you'll find that the lines turn out to be exactly the same. I want to show you one more method. There's more than one way to solve these, pick your points what you like. Let's do one more system of linear equations. Let's do 3x minus y is 9 and 2x plus y is 1. The addition method is when you look at this and say, you know what I can do to these equations, let me just add them. Usually this is a nice method to do when there are some obvious cancellation that's going to happen if you add or subtract. When I do the addition method and add this, I get 5x, the y is cancel is ten. You get a much simpler method. It's another easier way to eliminate a variable and then of course right from here I have x is 2. I don't want just x, I also want y. Now you pick any equation, first one or the second one, it does not matter and plug in so I plug into the first one I get 3 times 2 minus y is 9 or 6 minus y is 9. Subtract 6 and you get minus y is 6, y is minus 6. Your x coordinate is 2, your y coordinate is minus 6. This a nice way to do the additional method and it's a little bit of a shortcut. Once you have your value for x go ahead and plug it into any equation you want. It does not matter. Remember this got to satisfy both so pick the easiest one you want. If you plug it into the first one or you plug it in the second one, you can check, but you get y equals negative 3. Go ahead and check that these points do in fact satisfy the system. We have a nice consistent system. Again in your head you want to think about the x,y plane with two lines looking like an x, they cross and they meet at a single point. Let's do one more. What if I had 2x minus 3y is negative 2 and 3x minus 2y is 12. I say, well, the addition method here may not apply. I don't have some immediately obvious thing that if I started adding or subtracting would cancel and that's true. Maybe substitution would be better here but I just want to show you, you can do something to make these things cancel if you want. I can certainly multiply the top equation by three and I could certainly multiply the bottom equation by two. You are allowed to do any algebra you want as long as you don't break any rules. Whatever I do to the left side, I must do to the right side. Don't forget to multiply the right side by three and multiply the left side by minus 2. When you rewrite those systems, we then get a new system of equations. They're equivalent, but you get 6x minus 9y is negative 6 and the second equation becomes minus 6x plus 4y equals minus 24. Now it's more obvious that I should add these together and use the addition method. When you do that, the six and the negative 6 cancel. You get minus 5y plus 4y, that's minus 5y and then you get negative 6 minus 24, that's negative 30 and of course that means that y is 6. Once you have y is 6, pick any equation, first one second one, the revise one, either one you want plug in and you leave it to you to check me but you get x equals 8. Any equation you want, go ahead and pick another one. Your point to your coordinate here is 8,6. That is the solution to the system of linear equations. Let's do one word problem. Joseph made $25,000 profit on the sale of his condominium. He lent part of the profits to Nicholas at 10 percent interest and the remainder to Orlando at 8 percent. If he receives $2,200 in interest after one year, then how much was lent to each business. Take a second, pause the video, try to work this out so we're all on the same page. We're going to need to introduce our own variables here. They're asking how much was lent to each business. Let's call x the dollar amount lent to Nicholas and we'll let y be the dollar amount lent to Orlando. Now, this question is saying solve for x, solve for y. Try to set up the equations that correspond to this word problem. What's the first equation? I know that I lent some money to Nicholas and I know that I lent some money to Orlando so x plus y is $25,000 and I also know that my interest received $2,200 was a 10 percent. Careful 10 percent is.10 or just.1, I guess is fine, of what I lent to Nicholas and 8 percent.08 times y. From here I have a nice system of equations. To solve the system, it does not matter which method you pick. For no good reason I'm going to pick the elimination method, so I'll multiply the top by negative.1. When I do that I get negative.1x minus.1y equals negative, remember this is just 10 percent, 2500 and I'll keep that with the second equation, positive.1x plus.08y equals 2200. Now I'm going to add the equations and the reason why I do that is to eliminate the x. You get zero for x, I get.08 minus.1. That is negative.02y to negative 300. Divide by negative.02 and you get y is 15,000. Remember what this means is word problems so we're mindful of the units. This is $15,000. If the total numbers have to add to 25, I can take y and plug it back in and that gives me that x must be $10,000. We can answer the question saying that Nicholas was lent $10,000 and Orlando was lent $15,000. Go back and check. Make sure these numbers solve the system of equations. Great job on this word problem. Great job and all the algebra graphing that went into solving the systems. We'll see more applications and look at inequalities in the next video.
