Two-variable linear equations intro

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 2x minus 3 this is a linear equation now why do we call it a linear equation well if you were to take the set of all 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 let's let's actually feel good about that statement let's see if let's plot some of the XY XY pairs that satisfy this equation and then feel good that it does indeed generate a line so I'm just going to pick some X values that make it easy to calculate the corresponding Y values so if X is equal to 0 you're going to Y is going to be 2 times 0 minus 3 which is negative 3 and on our coordinate plane here that's we're going to move 0 in the X direction 0 in the horizontal direction we're going to go down 3 in the vertical direction in the Y direction so that's that point there if X is equal to 1 what is y equal 2 well 2 times 1 is 2 minus 3 is negative 1 so we move positive 1 in the X direction and negative 1 or down 1 in the Y direction now let's see if X is equal to 2 what is Y 2 times 2 is 4 minus 3 is 1 when X is equal to 2 y is equal to 1 and hopefully you're seeing now that if I were to keep going and I encourage you to if you want pause the video try x equals 3 or x equals negative 1 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 you will see the line that I'm talking about so let me see if I can draw I'm going to 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 2x minus 3 so if you were to graph all of the XY pairs that satisfy this equation you are going to get this line and you might be saying anyway 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 try to integer values of X but you could try any value in between here all of these it's actually a pretty neat 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 0.5 so if X is equal to negative 0.5 if we look at the line when X is equal to negative 0.5 it looks like it looks like Y is equal to negative 4 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 1/2 then what is y equal to let's see 2 times negative 1/2 I'll try it out 2 times negative 2 times negative 1/2 minus 3 well this is 2 times negative 1/2 is negative 1 minus 3 is indeed negative 4 it is indeed negative 4 so you can literally take any any for any x value that you put here in 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 just in a color you can see so this point represents a solution to a linear equation this point represents a solutions linear equation this point is not a solution to the linear equation so if X is equal to 5 then Y is not going to be equal to 3 if X is going to be equal to 5 you go to the line to see what the solution the linear equation is if X is 5 this shows us that Y is going to be 7 and it's indeed the fact that's indeed the case 2 times 5 is 10 minus 3 is 7 the point the point 5 comma 7 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's 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 4x minus 3y is equal to 12 this also is a linear equation and we can see that if we were to graph the XY pairs that satisfy this we get once again get a line x and y if X is equal to 0 then this goes away and you have negative 3y is equal to 12 see if negative 3y equals 12 then Y would be equal to negative 4 negative 0 comma negative 4 you can verify that 4 times 0 minus 3 times negative 4 well that's going to be equal to positive 12 and let's see if Y were to equal 0 if Y we're to equal 0 then this is going to be 4 times X is equal to 12 well then X is equal to 3 and so you have the point 0 comma negative 4 0 comma negative 4 on this line and you have the point 3 comma 0 on this line 3 comma 0 did I do that right yet so 0 comma negative 4 and then 3 comma 0 these are both going to be on this line 3 comma 0 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 X Y all the XY pairs that satisfy this if you were to plot them out it forms a line now what are some examples of maybe you're saying well the way we 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 nonlinear equations so a non nonlinear whoops need to 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 12 this is also not going to be a line or it could be something like 5 over X plus y is equal to 10 this also is not going to be a line so now at some point you could I encourage you to try to graph these things these are actually quite interesting but given that we've now seen examples of linear equations and nonlinear equations let's see if we can come up with a definition for linear equations one way to think about it is it's an equation that if you were to graph all 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 12 is a constant it's not going to change based on the value of some variable 12 is 12 or negative 3 is negative 3 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 this is just one why we're not dividing by X or Y we're not multiplying or 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 it's 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