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Systems of equations with substitution: y=-1/4x+100 & y=-1/4x+120

CCSS.Math: ,

Video transcript

we're given a system of equations here and we're told to solve for x and y now the easiest thing to do here since in both equations they're explicitly solved for y and say well if Y is equal to that and Y also has to equal this second equation then why don't we just set them equal to each other or another way to think about it is if Y is equal to this whole thing right over here that's what that first equation is telling us and if we have to find an x and y that satisfy both of these equations if y is equal to that Y I can I just substitute that right here for Y and if we do that this the left-hand side of this bottom equation becomes negative 1/4 X plus 100 and then that is going to be equal to this right hand side I'll do it in the same color is equal to negative 1/4 X plus 120 now the first thing we might want to do is maybe get all of our X terms on to the left or the right hand side of the equation and if we wanted to get rid of these X terms from the right hand side get them on the left hand side the best thing to do is to add 1/4 X to both sides of this equation so let me do that so we're going to add 1/4 X here add 1/4 X here and you might already be sensing that something shady is going on so let's do it so negative 1/4 X plus 1/4 X they cancel out you get 0 X so the left side of the equation is just to 100 and then the right side of the equation same thing negative 1/4 X plus 1/4 X they cancel out no X is and you're just left with is equal to 120 which we know cannot that is definitely not the case 100 is not equal to 120 we got this nonsensical equation here that 100 equals 120 so this type of system has no solution it has no solution and you know it has no solution because in order for it to have any solution these two numbers would have to be equal to each other and they are not equal to each other and if you look at the original equations it might jump out at you why they have no solutions both of these means or both of these equations if you view them as lines have the exact same slope they have the exact same slope but they have different y-intercepts so if I just were to do a really quick graph here if I were to do a really quick graph here that's my y-axis that is my x-axis it's Y and X this first graph over here it's y-intercept is 100 its y-intercept is let me do it a little bit lower its y-intercept let's say that that is 100 so it intersects right there and it's the slope of negative 1/4 so maybe it looks something like this maybe it looks something like that that's that first line this second line I'll do it in pink right here Y is equal to negative 1/4 X plus 120 its y-intercept might be right here at 120 but it has the same slope negative 1/4 so it's slope it's the line would look something like this so you see that there are no X&Y points that satisfy both of these equations another way to think about it if y is you take an X this first equation says ok you take your X multiply it by negative 1/4 and add 100 and that's going to give you Y now here we say well you take that same X and you multiply it by negative 1/4 and add 120 and that has to be equal to Y well the only way that that would ever be true is if 100-120 we're the same number and they're not the same number so you're never going to have a solution of this system these two lines are never going to intersect and that's because they have the exact same slope