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## Number of solutions to systems of equations

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# Number of solutions to a system of equations algebraically

CCSS Math: 8.EE.C.8, 8.EE.C.8a, 8.EE.C.8b, HSA.REI.D.10, HSA.REI.D.11

## Video transcript

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 And so it doesn't matter what x y values there are, You can never find an x y pair that's going to make 0 equal to negative 20. In fact the X's and Y's have
disappeared from this equation there's no way that this is going to be true so
we have no we have no solutions now if you were to plot these if you were to
plot each of these lines you would see that they are parallel lines and that's
why they have the same slope different y-intercepts and that's why we have no
solutions they don't intersect let's do another one of these this is this is fun
all right how many solutions does the following system of linear equations
have so let's do the same thing I'm going to keep the first equation the
same negative 6x plus 4y is equal to 2 and the second equation let me just see
if I can cancel out the X term so if I have a negative 6x if I multiply this by
2 I'm going to have a positive 6x so I can let's see I'm going to multiply this
whole equation both sides of it by 2 so I'm going to have 6x 3 x times 2 is 6x
negative 2y times 2 is negative 4y and that is going to be equal to negative 2
now let's do the same thing let's add the left sides and let's add the right
sides so negative 6x plus 6x well that's going to be 0 4y minus 4y that's 0 we
just have a 0 on the left-hand side now on the right-hand side we have 2 plus
negative 2 well that's 0 so this is a little bit different it still looks a
little bit bizarre 0 equals 0 last time we had 0 is equal to we add what 0 is
equal to negative 20 now we have 0 equals 0 so one way to think about it is
even though the X's and Y's are no longer in this equation okay well what
XY pairs is it going to be true for is that it's going to make it true that 0
is equal to 0 well this is going to be true no matter what x and y are in fact
x and y are not involved in this equation anymore 0 is always going to be
equal to 0 so this is going to have an infinitely this is going to have
infinitely many solutions here and that's because these are the same lines
they just look a little bit different algebraically but if you scale one of
them in the right way in fact if you just multiply both sides of this one the
second one by negative 2 you're going to get the top one and so they actually
represent the exact same lines you have an infinitely many solutions all right
when trying to find the solution to the following system of linear equations
Evon takes several correct steps that lead to the equation negative 5 is equal
to 20 how many solutions is the system of linear equations have I don't even
have to look at the system right over here the fact that she got the statement
that can never be true negative 5 is never going to be equal to 20 tells us
that she has no those solutions and once again if you
were to plot these graphically you would see that these are parallel lines that's
why they have no solutions they never intersect there's no XY pair that
satisfies both of these constraints let's do a let's do a couple more of
these when trying to find the solution to the following system of linear
equations Alba's take several correct steps that lead to the equation 5y is
equal to negative 5 and say how many solutions does this system of linear
equations have well 5y equals negative 5 we could divide both sides by 5 and we
get Y is equal to negative 1 and then if you substitute back in Y is equal to
negative 1 if you did it in this first equation if Y is equal to negative 1 all
of this becomes positive 2 you can subtract 2 from both sides and you get
5x is equal to 4 or you'd get what X is equal to 4/5 or if you put negative 1
over here you would get 5x minus 3 is equal to 1 you could add 3 to both sides
and you get 5x equals 4 again X is equal to 4/5 so you have exactly one solution
you would have X is equal to 4/5 y is equal to negative 1 let's do one more
when trying to find the solution of the following system of linear equations
Livan take several correct steps that lead to the equation 0 equals 0 so once
again I don't even need to look at this over here zero equals zero is always
going to be true so this is going to have an infinitely this is going to be
an infinitely many solutions