Intro to intercepts

let's say that we have the linear equation y is equal to 1/2 X 1/2 X minus 3 so if we wanted to draw the line that represents the set of all points all the coordinates where the x value and the y value satisfy this equation well we could start off by just trying to draw by trying to by trying to draw a few of those points and then connecting them with the line so let's set up a little table here X comma Y and then we could just try out a couple of X values here then figure out what the corresponding Y values are and I'm going to pick X values where's going to be fairly easy to calculate the Y values so let's say when X is equal to 0 then you're gonna have 1/2 times 0 minus 3 well then Y is going to be negative 3 when X is let's see let me try X is equal to 2 because then 1/2 times 2 is just going to be 1 so when X is equal to 2 you're going to have 1/2 times 2 is 1 minus 3 is negative 2 when X is equal to let's try 4 so 1/2 times 4 is 2 and then minus 3 is negative 1 and we can keep going but actually all we do need all we need is two points for line so we're ready to plot this line if we like the point 0 comma negative 3 is on this line 0 comma negative 3 actually let me do this in a slightly darker color so we can see it on this white background 0 comma negative 3 is on the line 2 comma negative 2 is on the line so 2 comma negative 2 and then we have 4 comma negative 1 so when X is 4 y is negative 1 and I could draw I could draw a line that connects that connects all of these so it would look something like hey let's see if I can do this so it would look something like it would look something like like that so this right over here this is literally this is the graph of y is equal to 1/2 X minus 3 now when we look at a graph like this an interesting thing that we might want to ask is where does the graph intersect our axes so first we could say well where does it intersect our x-axis and when you look at this it looks like it happens at this point right over here and this point where a graph intersects and axes this is called an intercept and this one in particular is called the x-intercept why is it called the x-intercept because that's where the graph is intersecting the x-axis and the x-intercept it looks like this is at the point 6 comma 0 now it's very interesting the x-intercept happens when y is equal to 0 remember you're on the x axis when you haven't moved up or down from the from that axis which means y is equal to 0 so your x-intercept happens at x equals 6 y equals 0 it's this coordinate now what about the y intercept well the y intercept is this point right over here this is where you intersect or I guess you could say intercept the y axis so this right over here that over there is the y intercept and the y intercept is that the coordinate that has a 0 for the x coordinate x is 0 here and Y is negative 3x is 0 and Y is negative 3 this was actually one of the points that were one of the the pairs that we first tried out and you can validate that 6 comma 0 satisfies this equation right over here if X is 6 1/2 times 6 is 3 minus 3 is indeed equal to 0 so now that we know what an x-intercept is it's the point where graph intersects the x axis or intercepts the x axis and the y intercept is the point where graph intercepts the y-axis or intersects the y axis let's try to see if we can find the x and y intercepts for a few other for a few other linear equations so let's say let's say that I had the linear equation let's say that I have 5x 5x plus 6y is equal to is equal to 30 encourage you to pause this video and figure out what are the x and y-intercepts for the graph that represents the solutions all the XY pairs that satisfy this equation well the easiest thing to do here let's see what what the what the y-value is when x is equal to 0 and what the x-value is when y is equal to 0 when x is equal to 0 this becomes 6 y equal 30 and so 6 times what is 30 well Y would be equal to 5 here so when x is 0 Y is 5 and what about when y is 0 when y is 0 that's going to be 0 and you have 5 X is equal to 30 well then X would be equal to 6 then X would be equal to 6 so we could plot those points 0 comma 5 when x is 0 Y is 5 and when and when X is 6 Y is 0 so those are both points on this graph and then the actual graph is going to or the actual line that represents the x and y pairs that satisfy this equation is going to look like it's going to look like this I'll just try so I can make it go it's going to look like it's going to go through those two points and so it's going to I can make it go the other way too you see it's going to go through those two points so it's going to look something like that now what are its x and y-intercepts well we already kind of figured it out both the intercepts themselves these are the points on the graph where they intersect the axes so this right over here this is the y intercept that point is the y intercept and it happens it's always going to happen when X is equal to 0 and when X is equal to 0 we know that Y is equal to 5 it's that point the point 0 comma 5 and what is the y intercept is the x intercept the x intercept is the point it's actually the same x-intercept for that other for this equation right over here it's the point 6 comma 0 that point right over there