how many solutions does the following system of linear equations have and 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 are
they parallel in which case they never intersect you'd have no solutions or do
they intersect exactly one place in case you would have 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'm going to leave that
unchanged 5x minus 9y is equal to is equal to is 16 now this second
equation right over here let's say I want to cancel out the X terms so let me
multiply the second equation by negative one so I have a negative 5x that I can
cancel out with the 5x so if I multiply the second equation by negative one I'm
going to have negative 5x plus 9y plus 9 Y is equal to negative 36 negative 36
now what I'm going to do is I'm going to add the left side of the equations
and the right sides of the equations to get a new equation so 5x minus 5x well
that's going to be 0 negative 9y plus 9y well that's going to be 0 again I don't
even have to write it it's just going to be 0 on the left-hand side and on
the right-hand side I'm going to have 16 minus 36 so negative 20 so now I'm
left with the somewhat bizarre-looking equation that says that 0 is equal to
negative 20 now one way you might say well how does this make any sense and
the way to think about it is well are there any X Y values 4 which is 0 is
going to be equal to negative 20 well no 0 is never going to be equal to negative
20 and so it doesn't matter what X Y values you can never find an X well X or
XY 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 Yvonne 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 Levon 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