Solution of Linear Equations - Matrices and Determinants, Business Mathematics & Statistics Video Lecture | Business Mathematics and Statistics - B Com

in this video I'm going to talk about
solving systems of linear equations
using substitution there's a bunch of
different ways you can solve systems of
linear equations and this is just one of
them so a couple things to talk about so
again your your linear equations well if
you graph them that just means you get
lines so there's kind of three
possibilities when you when you solve
these one thing that can happen
graphically is well your lines will
cross in one place and that means you
have exactly one solution algebraically
what's going to happen you'll be able to
solve for you'll get some number for X
you'll get some number for Y and that
will imply that you have exactly one
solution so if you get x equals
something y equals something that means
that they're crossing in one place
another possibility here is you know
you'll do some algebra and you'll end up
getting some sort of nonsensical
statement out like perhaps eight equals
zero well eight doesn't equal zero so
what that's going to imply is that your
two lines are actually parallel and in
that case they never cross and there's
not going to be any solution at all the
last case that can happen is well you go
to graph what looks like two different
lines they actually turn out to be the
same line and what's going to happen
algebraically is you'll get a statement
that says something like zero equals
zero and in that case it just means you
have infinitely many solutions so let's
do let's do a couple of these using
again substitution
suppose we have this system of equations
2x plus 4y equals 4 and y equals x minus
2 so what I'm going to do is since I
have the Y isolated already I'm going to
rewrite this as 2x plus 4 times y but
now I'm going to plug in the fact that Y
is equal to X minus 2 so I'll plug that
in there and then that's going to equal
4 so what this is going to do it's going
to allow me to solve this for X so I'll
get 2x plus 4x when I distribute I'll
get 4 times negative 2 which is negative
8 equals 4 2x + 4 X is 6x if I add 8 to
both sides
I'll get 12 on the other side and then
that'll tell me that x equals 2 so now
I've got one of my values and now to
find the other value I can take either
one of my two original equations and
simply plug in the fact that x equals 2
well it seems to me like the second
one's probably the easiest one to use so
if I plug in the fact that x equals 2
I'll get that Y is 2 minus 2 or that y
equals 0 so it says my solution is the
point 2 comma 0 and notice if you plug 2
0 into the first one you'll get 4 on the
left and if you plug 2 comma 0 into the
second one you'll get well you'll get 0
equals 0 if you go to graph these two
you'll find out that these are actually
crossing in one place namely at the
point 2 comma 0 so let's do another one
here X plus 3y equals 6 and 2x + 6 y
equals-- supposed to be in negative 12
I'm going to take my first equation here
and solve this for X seems like that
would be the easiest thing to do
x equals 6 minus 3y and now what I'm
going to do is I'm going to plug that
into the second equation I'll get 2
times the x value which is going to be 6
minus 3y and then I still have the plus
6y floating around equals negative 12 so
I'll get 12 minus 6y plus 6 y equals
negative 12 and that will give me looks
like 12 equals negative 12 well again
this is one of those kind of nonsensical
statements 12 doesn't equal negative 12
ever so what that simply says is is that
our original system of equations has no
solution
so if I do one other one here little
noise in the background there sorry for
that so let me do one other one here
I've got 2x minus 3y equals 6 and 4x
minus 6y equals 12
again you could solve for either one you
want
you know I guess I'll solve for the X
again that seems I don't know maybe
easier to me it seems pretty equivalent
so if I add 3y to both sides
I'll get 6 plus 3y I'm just rewriting my
first equation I'll divide divide both
sides by 2 so that'll give me x equals
remember you can when you have a single
thing in the bottom you divide each
thing by that number
so I'll get x equals 6 over 2 which is 3
plus three-halves times y and again what
I'm going to do is now plug that into
the other equation so I'll get 4 times X
again X is going to be this 3 plus 3
over 2 times y I still have a minus 6y
floating around equals 12 well if I
distribute on the left side I'll get 4
times 3 which is 12 if I distribute to
the other term I'll get 4 times 3 which
is 12 over 2 I'll get a positive 6y
minus 6y equals 12 and here my six Y's
are going to cancel out and I'm going to
get the statement that 12 equals 12 and
this is equivalent to what I was talking
to before if you get where you have
something equals something it turns out
that these are actually going to be the
same line and you're going to actually
have infinitely many solutions
okay and notice one thing that's
important to notice here notice if you
multiply this first equation by two by
two by two by two you'll get the second
one
if I multiply 2x by two I'll get four X
if I multiply negative three Y by two
I'll get negative 6y if I multiply six
by two I'll get 12 so if one line is a
multiple of the other it's going to turn
out that these are parallel lines so you
know this is one that once you kind of
get the hang of this you could just look
at this and say oh well this is really
the same line because if I multiply the
first one by two i'll get the second one
there's going to be infinitely many
solutions in this case alright so this
is the method of solving linear systems
of equations using substitution maybe
not necessarily the most straightforward
way but the idea is simple enough you
solve for X or you solve for y and one
of the equations you plug that into the
other one and then you get a value if
you once you get a value you plug it
into the other and again that's assuming
you have a single solution if you get a
statement like zero equals zero it means
you have infinitely many and if you get
a statement like you know pi equals 54
well that doesn't happen that means you
have no solution so there's also also
ways to do these using elimination by
addition you can actually write these in
a matrix form and I'll do some examples
of that in some other videos so if you
have some questions or comments I'm
always happy to hear them feel free to
shoot me an email