VIDEO-Systems of Equations

High class, this is miss champagne, and today I'm going to teach you about systems of equations. We've already worked with Y equals MX plus B equations previously. When we have two or more equations, we just call it a system, because both of these equations share a certain value. That means the same X that makes this equation true is the same number that makes this equation true. The same Y number that makes this equation true is the exact same Y number that makes this equation true. So let me explain to you what I mean. When we graph both of these lines on this equation, first, we look for. The last number. The last number is our Y intercept. So we have a positive three and a negative four. That is how we're going to be able to tell which line matches which equation. So I'm going to look for negative four on this Y axis. And when I go down here to negative four, I see that this is the line that matches. So this line goes with this first equation. Y equals three X minus four. When I look for the positive three on the Y axis, I see that this line is a line that matches this equation. Y equals negative one half X plus three. When these two lines meet at this point, this is their answer or solution. So go ahead and write. You can go ahead and write that this is their solution. The solution is the same thing as the answer. The solution is located at two, two. So here at the bottom, that's why you see two two. Because that is the solution to this system of equations. And remember a system just means two or more equations. Previously, we only work with one equation at a time. Now we're working with two equations. We can find the solution by looking at a graph, we can also find the solution by doing this mathematically. So let's look at how we would find two two mathematically. If you look here. We have bring it a little bit closer. All right, so we have a dollar bill. And we have four quarters. So all this is saying is that. This equation, that's what the with this represents. This equation is equal to a dollar bill. This is also the Y equation. This is equal to four quarters. What do we know about a dollar bill and four quarters? Yes, they're equal to each other. Four quarters is equal to $1. So this equation three X minus four represents a dollar bill. Negative one half X plus three represents four quarters. So what we're going to do is we're going to equal them to each other. So here, I'm going to put three X minus four is equal to negative one half X plus three. And then the first thing I'm going to do is I'm going to get. All. X's on one side. Of the equal sign. And all numbers. With no letters. On the other side. Of the equal sign. Bye. Doing these. You guessed it. By doing the opposite. So I'm going to rewrite it again. And I'm going to start doing the opposite. And you can start with any number. It doesn't matter which number you start out with. I see that there's a negative four here. I'm going to make this, I'm going to draw my wall. I'm going to make this a positive four. So I can do the opposite here. And then I'm going to find the number that's like it. This is a three with no letter behind it. So I'm going to write a positive four here. Whatever you write here, you have to go on the other side of this wall and write the exact same thing. Everything else, we're just going to bring it down. I'm going to circle my negative sign so that I don't forget it. I'm going to bring down the three X and negative four in a positive four equals zero, so you don't have to write anything here. Just bring down your negative one half and your X do not forget your ex. Then we have three plus four. Which is 7. Now the only two things that are left, we have an X here, and we have an X here. If I try to do the opposite of this three X because there's an imaginary plus sign in front of the three X so in order to get rid of it, I would have to do minus three X here. If I took this three X away, that means that there will be nothing on this side. We can not move this positive three X because we can not leave this side empty. So I need to write that. We can not move three X. Because. That side will be. Empty. With no numbers. And we can not leave. Any side. Empty. So we can not leave this item. If I can not move the three X over here, then I'm going to need to move this negative one half X over here. So I'm going to do the opposite of it. And I'm going to do plus one half X here. Plus one half X here. And then at positive 7, I'm just going to bring it down. So what I'm going to have a three X plus one half X is equal to that positive 7, because the negative one half X and the positive one half X equals zero. So you write nothing here. In order to add numbers, whole number and a fraction, you have to first put your imaginary one here. In order to add fractions, the denominators have to be the same. In order to add or. Subtract fractions. The bottom number. Called your denominator. Has to be. The same number. So I'm going to flip this paper over. So that. We can add the fractions. So again, we have three over one, X. Minus plus one half X equals 7. So in order to get these denominators the same, if this is a one and this is a two, that's easy. All you got to do is make these bottom numbers a number two. In order to make this one into a number two, what do we have to do to that number one? We have to multiply it by two. Whatever you do to the bottom, you got to do the same thing to the top. Three times two is 6. So now I have 6 over two X and what do I need to do to this to keep it a two? You just multiply it by one times one times one. So this stays the same. Equals 7. Once you have your light denominators, it took us a long time to work it out to get there to get this too. We're going to leave it the same. So leave same, which means do not add. Your bottom number. But you can add your top number. So we're going to add an in depth with 7 over two X is equal to 7. And we're going to make this into a fraction as well. I'm going to keep this X with the numerator. Remember when you have an equal sign between two fractions, that means that you can cross the road. So. An equal sign. Between two fractions. Means. That I can cross and I draw the lines and make it look like a road. So make the equal sign look like a road. I can cross the road. Which means cross. Multiply. So I can cross, multiply. So now I have 7 X times one, touch your equal sign, and you're going to cross the road. The other way, and you have a two times 7, two times 7. 7 times one, that's 7 X please don't forget to get your brain down your variable, two times 7 is 14. The last step, any time you have a number and letter side by side, there is an imaginary multiplication symbol in between. We're going to do the opposite. And we're going to divide both sides by 7. 7 divided by 7 is one. So we have X is equal to 14 divided by 7 is two. So we have our X value, the second step is to find the Y value. In order to find the Y value, we are going to take either one of these equations. You can choose whichever one it doesn't matter. And you're going to plug the X value back in so that we can find out what the Y value means. It's easier to work with whole numbers rather than the fraction. So we're going to plug the two back in here and see what Y equals. So we're going to write the equation Y equals three X minus four. And it's called evaluate. So at this point, we're going to evaluate. Evaluate means plug in the number. So we're going to evaluate Y equals three X minus four. We're going to plug in the number for X so I'm going to draw an empty box for X minus four. We just found out that X equals two. So we're going to put a two here in this box. A number and a letter side by side, remember there's an imaginary multiplication assembled in between each one. So we have Y is equal to three times two, which is 6. Minus four. So Y is equal to 6 minus four is two. So now we have our X value, and we have our Y value. So our solution. Or our answer is X is equal to two. Y is equal to two. So either you could do this mathematically like we just did by equaling the equations to each other or you could have easily looked at the graph to see where the two black lines touched or meet. At two comma two. Thank you.