in this video we are going to discuss linear models and systems of linear equations with a focus on a review of lines we are going to look at plotting ordered pairs in the cartesian coordinate system this covers page 17 and page 18 of your lecture notes first of all the cartesian coordinate plane is a grid system where you have two perpendicular axes the horizontal axis is called the x-axis and if we draw our axis this would be our horizontal axis which we could label as the x-axis then next we have the y-axis and often that is our vertical axis in this course this is our y-axis where the two meet so where these two axes meet that intersection is called the origin and has a value of 0 0. one thing that we want to note is when it comes to each of our axes as we move along our axis the negative numbers are on the left side of the axis and the lower part of the axis the positive numbers are on the right side and the upper part of the axis again that origin is 0 0 and all points on our cartesian plane so if i call this the point p it will have both an x coordinate and a y coordinate the x-coordinate is the distance from the origin so i could say that this is a negative a because it's to the left and so that will be my x-coordinate and my y-coordinate since it's above this would be a positive b giving me the ordered pair negative a positive b when we go to graph we will look for ordered pairs that we can plot on our cartesian plane and often when we do so we want to choose what we call nice numbers so that our equations we end up with integer values for a and b in our first example we would like to graph x minus two-fifths y equals negative one by plotting some points i have my cartesian plane here but i'm going to begin by creating a table of points when i think about my points i would like to have nice numbers so instead of substituting numbers in for x i'm going to substitute for y the reason i want to substitute for y is so that i can get rid of my fraction i would like to eliminate my fraction so i'm going to choose values for y i'm going to substitute them in so really i could rewrite my equation as x equals two-fifths y minus one and i'm going to substitute into the y to find my x and then i will get my ordered pair that i will later graph when i think of nice numbers my goal is to make two-fifths times some number give me an integer well zero is always great because if i substitute in zero the two-fifths times zero will be zero minus 1 which gives me negative 1 so my ordered pair is negative 1 0 remember because this represents my x value my y value was here but when we write our ordered pair it's always x comma y next if i think of negative 5 negative 5 will allow me to divide out my 5 in the denominator so two-fifths times negative five minus one well two-fifths times negative five gives me negative two minus one gives me negative three and i will have the ordered pair negative 3 negative 5 that i can also plot and while we only need two points in order to make a line just to check i'm going to graph a third point just to be on the safe side so i have two-fifths times five well again the fives will divide out leaving me with two two minus one gives me one i will have one 5 as my next ordered pair let's turn to our graph and actually plot our points as you can see i have my three ordered pairs plotted on my graph and now i'm going to draw the line through them my line may not be exactly straight on this board but for you you can use a straight edge or something in order to guarantee that you get a nice straight line but there would be the representation of x minus two fifths y equals negative one i like to write the equation of the line next to the line to keep track of which line i have just graphed now i have my nice pretty graph but i would like to take a look and see do i get that same line if i use technology and so when i use technology that requires me instead of solving for x i have to solve for y so we want to verify that y equals 5 halves x plus five halves is the same as the original equation that we started with we're going to go to our calculator on your calculator you are going to begin by going to the y equals button which should be at the top of your calculator right below the screen once you get to your y equals screen we need to enter our equation notice that these are all y equals that is why we need to rearrange our equation so we will put parentheses 5 halves the x is next to the alpha key and then we have plus five halves and one thing that i would strongly recommend is you put parentheses around your fractions to guarantee that what you're dividing by is not the 2x but just the 2 especially with that first term then we will press our graph button and if your calculator has not been used in a while it will graph on a standard 10 by 10 window if it has and you can't see your graph or it's outside of it all you need to do is go to zoom and you will go to option 6 for most of you which is your standard 10 by 10 window and it will reset your calculator to that standard window of 10 by 10. we can see that we get the same graph through negative 1 0 1 is it 5 and negative 3 negative 5 are all on my graph so i get the same result using my calculator if i solve for y