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CCSS.Math:

so we have a system of two linear equations here this first equation X minus 4y is equal to negative 18 and the second equation negative x plus 3y is equal to 11 now what we're going to do is find an x and y pair that satisfies both of these equations that's what solving the system actually means as you might already have seen there's a bunch of x and y pairs that satisfy this first equation in fact if you were to graph them they would form a line and there's a bunch of other x and y pairs that satisfy this other equation the second equation and if you were to graph them it would form a line and so if you find the x and y pairs that satisfy both that would be the intersection of the lines so let's do that so actually I'm just going to rewrite the first equation over here so I'm going to write X minus 4y is equal to negative 18 so we've already seen in algebra that as long as we do the same thing to both sides of the equation we can maintain our equality so what if we were to add and our goal here is to eliminate one of the variables so we have one equation with one unknown so what if we were to add this negative X plus 3y to the left-hand side here so negative x plus 3y well that looks pretty good because an X and a negative X are going to cancel out and we are going to be left with negative 4y plus 3y well that's just going to be negative Y so by adding the left-hand side of this bottom equation to the left-hand side of the top equation we were able to cancel out the exits we had X and we had a negative X that was very nice for us so what do we do on the right-hand side we've already said that we have to add the same thing to both sides of an equation we might be tempted we might be tempted to just say well if I have to add the same thing to both sides well maybe I have to maybe have to add a negative x plus 3y to that side but that's not going to help us much we're going to have negative 18 minus X plus 3y we won't we would have introduced an X on the right-hand side of the equation but what if we could add something that's equivalent to negative x plus 3y that does not introduce the X variable well we know that the number eleven is equivalent to negative x plus three why why do we how do we know that well that second equation tells us that so once again all I'm doing is I'm adding the same thing to both sides of that top equation on the Left I'm expressing it as negative x plus 3y but the second equation tells us that that negative x plus 3y is going to be equal to eleven it's introducing that second that second constraint and so let's add 11 to the right-hand side which is what's it I know I keep repeating it's the same thing as negative x plus 3y so negative 18 plus 11 is negative seven and since we added the same thing to both sides the Equality still holds and we get negative Y is equal to negative seven or divide both sides by negative 1 or multiply both sides by negative one so multiply both sides by negative one we get Y is equal to seven so we have the y coordinate of the XY pair that satisfies the both of these now how do we find the X well we can just substitute this y equals seven to either one of these when y equals seven we should get the same x regardless of which equation we use so let's use the top equation so we know that X minus four times instead of writing Y I'm going to write four times seven because we're going to figure out what is X when y seven that is going to be equal to negative eighteen and so let's see negative four or four times seven that is 28 so let's see I could to solve for X I can add 28 to both sides so add 28 to both sides on the left hand side negative 28 positive 20 those cancel out I'm just left with an X and on the right hand side I get negative eight negative 18 plus 28 is 10 so there you have it I have the XY pairs that satisfy here that satisfies both x equals 10 y equals 7 I could write it here so I could write it as coordinates I could write it as 10 comma 7 and notice what I just did here I encourage you to substitute y equals 7 here and you will also get x equals 10 either way you would have you would have come to x equals 10 and to visualize what is going on here let's visualize it really fast let me draw some coordinate axes whoops I'm going to draw a straighter line than that all right there you go so let's say that is our y-axis and that is that is whoops that is our x-axis and then let's see this the top lot the top equation is going to look something like this it's going to look something like this and then that bottom equation is going to look something something like let me draw it a little bit nicer than that it's going to look something like this something like that we draw the bottom one here so you see the point of intersection and so the point of intersection right over here that is an XY pair that satisfies both of these equations and that we just saw it happens when X is equal to 10 and Y is equal to 7 once again this white line that's all the x and y pairs that satisfy the top equation this orange line that's all the X and y pairs that satisfy the orange equation and where they intersect that point is on both lines it satisfies both equations and once again take x equals 10 y equals 7 substitute it back into either one of these and you will see that it holds