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# Systems of equations with graphing: exact & approximate solutions

## Video transcript

the following two equations form a linear system all right this is one equation it has X&Y so it's going to define a line and then I have another equation that involves X&Y so it's going to define another line and it says graph the system of equations and find its solution so we're going to try to find it visually so let's graph this first one and to graph this line I have a little graphing tool here notice I can if I can figure out two points I can move those points around and it's going to define our line for us so let's pick I'm going to pick two X values and figure out the corresponding Y values and then graph the line so let's let's see how I could do this so let's see the easy one is what happens when x is equal to zero well if X is equal to zero everything I just shaded goes away and we're left with negative three Y is equal to nine so negative three y equals nine Y would be negative three so when X is equal to zero Y would be negative three so let me graph that when X is equal to zero Y X is zero Y is negative three now another easy point actually instead of trying another x value let's think about when Y is equal to zero because this is these equations are in standard form so it's easy to just test well what are the x and y intercepts so when y is equal to zero when y is equal to zero this term goes away and you have negative x is equal to 9 or X would be equal to negative 9 so when y is 0 x is negative 9 so when y is 0 X is negative 9 or when X is negative 9 Y is 0 so I've just plotted I've just plotted this first equation so now let's do the second one we'll do the same thing what happens when X is equal to zero when X is equal to 0 so this is going to be this is going to be our y intercept now when X is equal to 0 negative 6 Y is equal to negative 6 well Y would have to be equal to 1 so when x is 0 Y is equal to 1 so when x is 0 x is 0 Y is equal to 1 get one more point here when y is 0 when this term z-y being zero to make this entire term zero then 6x is equal to negative 6 or X is equal to negative 1 so when y is zero X is negative 1 or when X is negative 1 Y is 0 when X is negative 1 Y is 0 and so just like that I've plotted the two lines and the solution to the system is the are the x and y values that satisfy both equations and if they satisfy both equations that means they sit on both lines and so in order to be on both lines they're going to be at the point of intersection and I see this point of intersection right over here it looks pretty clear that this is the point X is equal to negative 3 and Y is equal to negative 2 so it's the point negative 3 comma negative 2 so let me write that down negative 3 comma negative 2 and then I could I could check my answer got it right let's do another let's do another one of these may be of a different type so over here it says the system of two linear equations is graphed below approximate the solution of the system alright so here I just have to just look at this carefully and think about where this point is so let's think about first its x value so it's x value let's see it's about right there in terms of its x value it looks like so this is negative 1 negative this is negative 2 so negative 1.5 is going to be right over here it's a little bit to the left of negative 1.5 so it's even more negative I'd say negative 1.6 so when I'm approximating it negative 1.6 hopefully it has a little leeway now how it checks the answer what about the Y value so if I look at the Y value here it looks like it's a little less than one and a half one and a half would be halfway between one and two it looks like it's a little less than halfway between one and two so I don't know I'd give it one point four positive one point four and let's let's check the answer see see how we're doing yeah we we got it right let's do let's actually just do one more for for good measure so this is another system we've just written the equations in more of our slope-intercept form so see Y is equal to negative 7x plus 3 when X is equal to 0 we have our y-intercept y is equal to 3 so when X is equal to 0 y is equal to 3 and then we see that our slope is negative 7 when you increase X by 1 you decrease Y by 7 so when you increase X by 1 you decrease Y by 1 2 3 4 5 6 & 7 when X goes from 0 to 1 Y went from 3 to negative 4 it went down by 7 so that's that first one now the second one our y-intercept when X is equal to 0 Y is negative 3 so let me graph that when x is 0 Y is equal to negative 3 and then its slope is negative 1 when x increases by 1 Y decreases by 1 so the slope here is negative 1 so when x increases by 1 Y decreases by 1 and there you have it you have your point of intersection you have the XY pair that satisfies both equations that is the point of intersection it's going to sit on both lines which is why it's the point of intersection and that's the point x equals 1 y is equal to negative 4 so you have x equals 1 and y is equal to negative 4 and I can check my answer and we got it right