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Current time:0:00Total duration:5:10

Solutions to systems of equations: dependent vs. independent

Video transcript

is the system of linear equations below dependent or independent and they give us two equations right here before I tackle this specific problem let's just do a little bit of review of what dependent or independent means and actually I'll compare that to consistent and inconsistent so just to start off with if we're dealing with systems of linear equations in two dimensions there's only three possibilities that these that the lines or the equations can have relative to each other so let me draw the three possibility so let me draw three coordinate axes so that's my first xx at x-axis and y-axis so x and y let me draw another one that is X and that is y let me draw one more because there's only three possibilities in two dimensions x and y if we're dealing with linear equations x and y so you can have the situation where the lines just intersect in one point we do this in so you could have one line like that and maybe the other line does something like that and they intersect at one point you could have the situation where the two lines are parallel so you could have a situation actually let me draw it over here where you have one line that goes like that and then the other line has the same slope but it's shifted it has a different y-intercept so maybe it looks like this and you have no points of intersection and then you could have the situation where they're actually the same line so that both lines have the same slope and the same y intercept so really they are the same line they intersect on an infinite number of points every point on either of those lines is also point on the other line so just to give you a little bit of the terminology here and we learned this in the last video this type of this type of system where they don't intersect where you have no solutions this is an inconsistent inconsistent system inconsistent system and by definition or I guess just taking the opposite of inconsistent both of these would be considered consistent both of these are consistent but then with inconsistent there's obviously a difference here we only have one solution these are two different lines intersect in one place and here they're essentially the same exact line and so we differentiate between these two scenarios by calling this one over here independent independent and this one over here dependent dependent so independent both lines are doing their say their own thing they're not they're not dependent on each other they're not the same line they will intersect at one place dependent they're the exact same line any point that satisfies one line will satisfy the other any point that satisfies one equation will satisfy the other so with that said let's see if this system this linear the system of linear equations right here is dependent or independent so they're kind of having this assume that it's going to be consistent that we're either going to intersect in one place we're going to intersect in an infinite number of places and the easiest way to do this we already have this second equation here it's already in slope-intercept form we had we know the slope is negative two the y-intercept is eight let's put this first equation let's put this first equation up here in slope intercept form and see if it has a different slope or a different intercept or maybe it's the same line so we have 4x plus 2y is equal to 16 we can subtract 4x from both sides we what we want to do is isolate the Y on the left-hand side so let's subtract 4x from both sides the left-hand side we are just left with a 2y and then the right-hand side we have a negative 4x plus 16 I just wrote the negative 4 in front of the 16 just so that we have it in the traditional slope-intercept form and now we can divide both sides of this equation by 2 so that we can isolate the Y on the left-hand side divide both sides by 2 we are left with Y is equal to negative 4 divided by 2 is negative 2x plus 16 over 2 plus 8 so all I did is algebraically manipulated this top equation up here and when I did that when I solved essentially for y I got this right over here which is the exact same thing as the second equation we have the exact same slope negative 2 negative 2 and we have the exact same y-intercept 8 & 8 if I were to graph these equations if I were to graph these equations that's my x-axis and that is my y-axis both of them have a y-intercept at 8:00 and then have a slope of negative 2 so they look something I'm just drawing an approximation of it but they look something they would look something like that so maybe this is the graph of this equation right here this first equation and then the second equation will be the exact same graph it has the exact same y-intercept and the exact same slope so clearly these two lines are dependent dependent they have an infinite number of points that are common to both of them because they're the same line