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# Number of solutions to a system of equations algebraically

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how many solutions does the following system of linear equations have i have my system right over here there's a couple of ways to think about it one way is to think about them graphically and think about well, are they the same line in which case they would have an infinite number of solutions or they are parallel in which case they never intersect you would have no solution or they intersect exactly at one place, in that case, you would have exactly one solution but instead, we're going to do this algebraically So let's try to actually just solve the system and see what we get So, the first equation...I will leave that unchanged 5x-9y=16 Now this second equation right over here, let's say I wanna cancel out the x terms so let me multiply the second equation by negative 1 so I have a -5x that I can cancel out with the 5x so if I multiply this second equation by -1 we will have -5x + 9y = -36 Now I'm going to add to the left side of the equation and the right side of the equation to get a new equation so 5x - 5x well that's going to make a zero -9y + 9y well that's gonna be zero again i don't even have to write it, it's gonna be zero on the left side and on the right-hand side I'm gonna have 16 - 36 = -20 So now I'm left with the somewhat bizarre looking equation that says that 0 is equal to -20 Now one way, you might say, "Well-well how does this make any sense?" And the way to think about it is: "Well are there any x y values for which 0 is going to be equal to -20?" Well no, 0 is *never* going to be equal to -20