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we're told to solve and graph the solution for the system of equations right here and the first thing that jumps out at me is that we might be able to eliminate one of the variables and if we just focus on the X we have a 4x here and we have a 2x right here if we were just add them right now we would get a 6x so that wouldn't eliminate it but if we can multiply this 2 X by negative 2 it'll become a negative 4x and then when you add it they would cancel out so let's multiply this equation the second equation by negative 2 so I'm going to multiply both sides of this equation by negative 2 and the whole motivation is so that this 2x becomes a negative 4x and of course I can't just multiply only the 2x anything I do to the left hand side of the equation I have to do to every term and I have to do to both sides of the equation so the second equation becomes negative 4x that's negative 2 times 2x plus we have negative 2 times negative y which is plus 2i is equal to 2.5 times negative 2 is equal to negative 5 I just rewrote the second equation multiplying both sides by negative 2 now this top equation I'll write it on the bottom now we have 4x minus 2y is equal to positive 5 and now we can eliminate we can say hey look the negative 4x and the positive 4x should cancel out or they will cancel out so let's add these two equations let's add the left side to the left side the right side to the right side and we can do that because these two things are equal we're doing the same thing to both sides of the equation so what do we get if we take our negative 4x plus our 4x well those cancel out so you're left with nothing maybe I can write a zero there 0x if you want and then you have your plus 2y and your negative 2i those also cancel out so you're also left with 0y and then that equals negative 5 plus 5 is equal to 0 so you have just have this 2 just simplifies to 0 equals 0 which is true but it's kind of bizarre we had all these X's and Y's everything everything canceled out so let's explore this a little bit more let's graph it and see what this is 0 equals 0 is telling us when we tried to solve the system of equations so let me graph let me graph this top guy I'll do it in blue so right now it's in standard let's put it in slope intercept form so we have 4x minus 2y is equal to 5 let's subtract 4x from both sides subtract 4x I want the X terms on the right-hand side so that I'm left with negative 2y is equal to negative 4x plus 5 now we can divide both sides by negative 2 by negative 2 and we are left with Y is equal to is equal to positive 2x right that's positive 2x 4 minus 2 point 5 minus 2 point 5 so let's graph that the y-intercept is negative 2 & a half so negative 2 and 1/2 right there and then it has a slope of 2 so if we move up 1 we go if we move up in the X direction well if we move to the right one in the positive x-direction we will move up 2 so 1/2 right there and if we were to do it again we move up 1 2 just like that so the line is going to look something like this try my best to draw a straight line this is the hardest part about a lot of these problems there you go so that's the top equation now let me draw the bottom equation let me draw and I'll do it in in this green color so this bottom equation was 2x minus y is equal to 2.5 and we can subtract 2x from both sides let's subtract 2x from both sides the left-hand side becomes negative y is equal to 2x plus or is equal to negative 2x plus 2.5 now let's multiply or divide both sides by negative 1 and you get Y is equal to positive 2x minus 2.5 and let's try to graph this and you already might notice something interesting about these two equations you try to graph this the y-intercept is at negative two point five right there the slope is two so it's going to be this exact same line it's going to be this exact same line and you saw that algebraically I didn't have to graph it these two lines have the exact same equation when you put them when you put them in slope intercept form that's the first equation that's the second equation so with the zero equals zero is telling us is actually that these are the same line that these actually have an infinite number of solutions any point on this line which is both of those lines will satisfy both of these equations you give me an arbitrary y solve for x in the top equation that that x and y will also satisfy the bottom equation so this actually has an infinite number of solutions these are the same line