Video Transcript: Two-Variable Equations

Two-variable linear equations intro 

[Voiceover] What I'd like to introduce you to in this video is the idea of a Linear Equation. And  just to start ourselves out, let's look at some examples of linear equations. So, for example the  equation y is equal to two x minus three, this is a linear equation. Now why do we call it a linear  equation? Well if you were to take the set of all of the xy pairs that satisfy this equation and if  you were to graph them on the coordinate plane, you would actually get a line. That's why it's  called a linear equation. And let's actually feel good about that statement. Let's see, let's plot  some of the xy pairs that satisfy this equation and then feel good that it does indeed generate a line. So, I'm just gonna pick some x values and make it easy to calculate the corresponding y  values. So, if x is equal to zero y is gonna be two times zero minus three which is negative  three. And on our coordinate plane here that's-- we're gonna move zero in the x direction, zero  in the horizontal direction and we're gonna go down three in the vertical direction, in the y  direction. So, that's that point there. If x is equal to one, what is y equal to? Well two times one  is two, minus three is negative one. So we move positive one in the x direction and negative  one, or down one, in the y direction. Now let's see, if x is equal to two what is y? Two times two  is four, minus three is one. When x is equal to two y is equal to one. And hopefully you're seeing  now that if I were to keep going, and I encourage you though if you want pause the video and  try x equals three or x equals negative one and keep going. You will see that this is going to  generate a line. And in fact, let me connect these dots and you will see the line that I'm talking  about. So, let me see if I can draw, I'm gonna use the line tool here. Try to connect the dots as  neatly as I can. There you go. This line that I have just drawn, this is the graph, this is the graph  of y is equal to two x minus three. So if you were to graph all of the xy pairs that satisfy this  equation you are gonna get this line. And you might be saying, "Hey wait wait, hold on Sal, you  just tried some particular points, why don't I just get a bunch of points, how do I actually get a  line?" Well, I just tried, over here I just tried integer values of x. But you can try any value in  between here, all of these, it's actually a pretty unique concept. Any value of x that you input  into this, you find the corresponding value for y, it will sit on this line. So for example, for  example, if we were to take x is equal to, actually let's say x is equal to negative point five. So if  x is equal to negative point five if we look at the line when x is equal to negative point five it  looks like it looks like y is equal to negative four. And that looks like that sits on the line. Well  let's verify that. If x is equal to negative, I'll write that as negative one half, then what is y equal  to? Let's see, two times negative one half, I'll just write it out, two times negative-- two times  negative one half minus three. Well this says two times negative one half is negative one minus  three is indeed negative four it is indeed negative four. So you can literally take any, any-- for  any x value that you put here and the corresponding y value it is going to sit on the line. This  point right over here represents a solution to this linear equation. Let me do this in a color you  can see. So this point represents a solution to a linear equation. This point represents a solution  to a linear equation. This point is not a solution to a linear equation. So if ex is equal to five then  y is not gonna be equal to three. If x is gonna be equal to five you go to the line to see what the  solution to the linear equation is. If x is five this shows us that y is going to be seven. And it's  indeed-- that's indeed the case. Two times five is ten, minus three is seven. The point-- the  point five comma seven is on, or it satisfies this linear equation. So if you take all of the xy pairs  that satisfy it, you get a line. That is why it is called a linear equation. Now, this isn't the only 

way that we could write a linear equation. You could write a linear equation like-- let me do this  in a new color. You could write a linear equation like this: Four x minus three y is equal to  twelve. This also is a linear equation. And we can see that if we were to graph the xy pairs that  satisfy this we would once again get a line. X and y. If x is equal to zero, then this goes away and you have negative three y is equal to twelve. Let's see, if negative three y equals twelve then y  would be equal to negative four. Nega-- zero comma negative four. You can verify that. Four  times zero minus three times negative four well that's gonna be equal to positive twelve. And  let's see, if y were to equal zero, if y were to equal zero then this is gonna be four times x is  equal to twelve, well then x is equal to three. And so you have the point zero comma negative  four, zero comma negative four on this line, and you have the point three comma zero on this  line. Three comma zero. Did I do that right? Yep. So zero comma negative four and then three  comma zero. These are going to be on this line. Three comma zero is also on this line. So this is, this line is going to look something like-- something like, I'll just try to hand draw it. Something  like that. So once again, all of the xy-- all of the xy pairs that satisfy this, if you were to plot  them out it forms a line. Now what are some examples, maybe you're saying "Wait, wait, wait,  isn't any equation a linear equation?" And the simple answer is "No, not any equation is a linear  equation." I'll give you some examples of non-linear equations. So a non-- non-linear, whoops  let me write a little bit neater than that. Non-linear equations. Well, those could include  something like y is equal to x-squared. If you graph this you will see that this is going to be a  curve. it could be something like x times y is equal to twelve. This is also not going to be a line.  Or it could be something like five over x plus y is equal to ten. This also is not going to be a line.  So now, and at some point you could-- I encourage you to try to graph these things, they're  actually quite interesting. But given that we've now seen examples of linear equations and non linear equations, let's see if we can come up with a definition for linear equations. One way to  think about is it's an equation that if you were to graph all of the x and y pairs that satisfy this  equation, you'll get a line. And that's actually literally where the word linear equation comes  from. But another way to think about it is it's going to be an equation where every term is  either going to be a constant, so for example, twelve is a constant. It's not going to change  based on the value of some variable, twelve is twelve. Or negative three is negative three. So  every term is either going to be a constant or it's going to be a constant times a variable raised  to the first power. So this is the constant two times x to the first power. This is the variable y  raised to the first power. You could say that bceause this is just one y. We're not dividing by x or  y, we're not multiplying, we don't have a term that has x to the second power, or x to the third  power, or y to the fifth power. We just have y to the first power, we have x to the first power.  We're not multiplying x and y together like we did over here. So if every-- if every term in your  equation, on either side of the equation, is either a constant or its just some number times x,  just x to the first power or some number times y, and you're not multiplying your x's and y's  together you are dealing with a linear equation. 

Solutions to 2-variable equations 

Which of the ordered pairs is a solution of the following equation? Negative three x minus y is  equal to six. What we have to remind ourselves is when we're give an ordered pair, the first  number is the x coordinate and the second number is the y coordinate, or the y value. So when 

they tell us the ordered pair, negative four comma four, they're saying "hey look, if x is equal to  "negative four, and y is equal to positive four, "does that satisfy this equation?" And what we  can do, is we can just try that out. So we have negative three and everywhere we see an x,  everywhere we see an x, we can replace it with negative four. So it's negative three times  negative four, minus, minus and everywhere we see a y, we can replace it with positive four. We replace it with positive four. So negative three times x minus y, which is four, needs to be equal  to six. Needs to be equal to six. Now is this indeed the case? Negative three times negative four is positive 12. Positive 12 minus four, positive 12 minus four is equal to eight, it's not equal to  six. Is not equal, is not equal to six. So this one does not work out. So let's see, negative three  comma three. We can do the same thing here. Let's see what happens when x is equal to  negative three and y is equal to positive three. So we substitute back in, we get negative three.  Negative three times x, which now we're going to try out x being equal to negative three.  Minus y, minus y. Y is positive three here. Minus y, gonna do that y color blue. Minus y now  needs to be equal to, now needs to be equal, just like before needs to be equal to six. So  negative three times negative three. That's going to be positive nine. Nine minus three is  indeed equal to six. Nine minus three is indeed equal to six. Nine minus three is six. That is  equal to six. This works out. So negative three comma three is an ordered pair that is a solution  to this equation. 

Worked example: solutions to 2-variable equations 

"Which of the ordered pairs is a "solution of the following equation?" 4x minus one is equal to  3y plus five. Now, when we look at an ordered pair we wanna figure out whether it's a solution,  we just have to remind ourselves that in these ordered pairs the convention, the standard, is is  that the first coordinate is the x coordinate, and the second coordinate is the y coordinate. So  they're gonna, if this is a solution, if this ordered pair is a solution, that means that if x is equal  to three and y is equal to two, that that would satisfy this equation up here. So let's try that out. So, we have four times x. Well we're saying x needs to be equal to three, minus one, is going to  be equal to three times y. Well, if this ordered pair is a solution then y is going to be equal to  two, so three times y, y is two, plus five. Notice all I did is wherever I saw the x, I substituted it  with three, wherever I saw the y, I substituted it with two. Now let's see if this is true. Four  times three is twelve, minus one. Is this really the same thing as three times two which is six,  plus five? See, 12 minus one is 11, six plus five is also 11. This is true, 11 equals 11. This pair three, two does satisfy this equation. Now let's see whether this one does, two, three. So this is saying when x is equal to two, y would be equal to three for this equation. Let's see if that's true. So  four times x, we're now gonna see if when x is two, y can be three. So four times x, four times  two, minus one is equal to three times y, now y we're testing to see if it can be three. Three  times three plus five, let's see if this is true. Four times two is eight, minus one, is this equal to  three times three? So that's nine plus five. So is seven equal to 14? No, clearly seven is not equal to 14. So these things are not equal to each other. So this is not a solution, when x equals two y  cannot be to three and satisfy this equation. So only three, two is a solution. 

Completing solutions to 2-variable equations 

So this is an example from the Khan Academy exercise, graphing solutions to two variable  linear equations. And they tell us to complete the table so each row represents a solution of the

following equation. And they give us the equation, and then they want us to figure out, what  does y equal when x is equal to negative five? And what does x equal when y is equal to eight.  And to figure this out, I've actually copied and pasted this part of the problem onto my  scratchpad, so let me get that out. And so this is the exact same problem, there's a couple of  ways that we could try to tackle it. One way, is you could try to simplify this more, get all your  xs on one side and all your ys on the other side. Or we could just literally substitute when x  equals negative five, what must y equal? Actually, let me do it the second way, first. So if we  take this equation, and we substitute x with negative five, what do we get? We get negative  three times, well, we're gonna say x is negative five, times negative five, plus seven y is equal to five times, x is once again, it's gonna be negative five, x is negative five, five times negative five, plus two y. See, negative three times negative five is positive 15, plus seven y, is equal to  negative 25 plus two y. And now, to solve for y, let's see, I could subtract two y from both sides,  so that I get rid of the two y here on the right. So let me subtract two y, subtract two y from  both sides. And then if I want all my constants on the right hand side, I can subtract 15 from  both sides. So let me subtract 15 from both sides. And I'm going to be left with 15 minus 15,  that's zero, that's the whole point of subtracting 15 from both sides, so I get rid of this 15 here.  Seven y minus two y. Seven of something minus two of that same something is gonna be five  of that something. It's gonna be equal to five y, is equal to negative 25 minus 15. Well, that's  gonna be negative 40. And then two y minus two y, well, that's just gonna be zero. That was the whole point of subtracting two y from both sides. So you have five times y is equal to negative  40. Or, if we divide both sides by five, we divide both sides by five, we would get y is equal to  negative eight. So when x is equal to negative five, y is equal to negative eight. Y is equal to  negative eight. And actually we can fill that in. So this y is going to be equal to negative eight.  And now we gotta figure this out. What does x equal when y is positive eight? Well, we can go  back to our scratchpad here. And I'll take the same equation, but let's make y equal to positive  eight. So you have negative three x plus seven, now y is going to be eight, y is eight, seven  times eight is equal to five times x, plus two times, once again, y is eight, two times eight. So  we get negative three x plus 56, that's 56, is equal to five x plus 16. Now, if we wanna get all of  our constants on one side, and of all of our x terms on the other side, well, what could we do?  Let's see, we could add three x to both sides. That would get rid of all the xs on this side, and  put 'em all on this side. So we're gonna add three x to both sides. And, let's see, if we want to  get all the constants on the left hand side, we'd wanna get rid of the 16, so we could subtract 16 from the right hand side, if we do it from the right, we're gonna have to do it from the left as  well. And we're gonna be left with, these cancel out, 56 minus 16 is positive 40. And then, let's  see, 16 minus 16 is zero. Five x plus three x is equal to eight x. We get eight x is equal to 40. We  could divide both sides by eight, and we get, five is equal to x. So this right over here is going to be equal to five. So let's go back, let's go back, now. So when y is positive eight, x is positive  five. Now they ask us, "Use your two solutions "to graph the equation." So let's see if we can do, oh, whoops, let me, let me use my mouse now. So to graph the equations. So when x is  negative five, y is negative eight. So the point negative five comma negative eight. So that's  right over there. So let me move my browser up so you can see that. Negative five, when x is  negative five, y is negative eight. And when x is positive five, and we see that up here, when x is  positive five, y is positive eight. When x is positive five, y is positive eight. And we're done. We 

can check our answer, if we like. We got it right. Now, I said there was two ways to tackle it, I  kind of just did it, I guess you could say, the naive way. I just substituted negative five directly  into this and solved for y. And then I substituted y equals positive eight directly into this, and  

then solved for x. Another way that I could have done it, that actually probably would have  been, or, it would for sure, would have been the easier way to do it, is ahead of time to try to  simplify this expression. So what I could have done, right from the get-go, is said, "Hey, let's  put all my xs on one side, "and all my ys on the other side." So this is negative three x plus seven y is equal to five x plus two y. Now let's say I wanna get all my ys on the left and all my xs on the  right. So I don't want this negative three x on the left, so I'd wanna add three x. Adding three x  would cancel this out, but I can't just do it on the left hand side, I have to do it on the right hand  side as well. And then, if I wanna get rid of this two y on the right, I could subtract two y from  the right, but, of course, I'd also wanna do it from the left. And then what am I left with? So  negative three x plus three x is zero, seven y minus two y is five y. And then I have five x plus  three x is eight x. Two y minus two y is zero. And then if I wanted to, I could solve for y, I could  divide both sides by five and I'd get y is equal to 8/5 x. So, this right over here represents the  same exact equation as this over here, it's just written in a different way. All of the xy pairs that  satisfy this, would satisfy this, and vice versa. And this is much easier. Because if x is now  negative five, if x is negative five, y would be 8/5 times negative five, well, that's going to be  negative eight. And when y is equal to eight, well, you actually could even do this up here, you  could say five times eight is equal to eight x, and then you could see, well five times eight the  same thing as eight times five, so x would be equal to five. So I think this would actually have  been a simpler way to do it. You see it all, I was able to do the entire problem in this little white  space here, instead of having to do all of this, slightly, slightly hairier, algebra.