Let's do some more systems of equations problem.
In this video, we're going to encounter systems that might
have no solutions, or that might have an infinite many
solutions, and we'll label them with words.
So let's start with one.
Let's say we have 3x minus 4y is equal to 13.
Let's say my other equation in my system is y is equal to
negative 3x minus y.
So the first thing-- this is kind of in a strange form
right here.
I want to get into the standard form and maybe I'll
do elimination for this systems. Let me rewrite the
top equation.
We have 3x minus 4y is equal to 13.
And let me rearrange this bottom equation here.
So if I were to add 2y-- well, let me subtract y from both
sides of this equation.
So if I subtract y from both sides of this equation, it
becomes 0 is equal to negative 3x minus 2y, or negative 3x
minus 2y is equal to 0.
So let me write that over here.
And it looks nice because I have a negative 3x here, I
have a positive 3x here.
Looks well suited for elimination.
So, I have negative 3x minus 2y is equal to 0.
So let's add the left-hand side of this equation to the
left-hand side of the yellow equation.
And we're going to add 0 to the right-hand side of the
yellow equation and we're essentially
adding 0 to both sides.
We're adding the same quantity to both sides, which we can
always do with an equation.
So, the left hant-side, the 3x cancels out with a 3x and
we're left with negative 4y minus 2y.
You get negative 6y is equal to 13.
Divide both sides by negative 6.
We are left with y is equal to negative 13/6.
Now let's solve for x.
And we can solve for x using either of these equations.
Let's use that top one, just for fun.
So we have 3 times x minus 4 times negative
13/6 is equal to 13.
Now we have a negative times a negative, so those are both
going to become positives.
And then the 4/6, that's the same thing as 2/3.
So this becomes 3x plus 2 times 13 which is 26/3, is
equal to 13.
Instead of 13, since I'm about to subtract 26/3 from both
sides, let me rewrite 13 as 39/3.
right?
39 divided by 3 is 13.
So, let me subtract 26/3 from both sides.
The left-hand side becomes 3x-- these cancel out-- is
equal to 39 minus 26 is 13/3.
And then we're going to want to divide both sides by 3.
Or you can view it as multiplying both sides by 1/3.
The left-hand side, we're just left with an x.
The right-hand side, x is going to be equal to 13/9.
So this system had a well defined solution.
The solution is x is equal to 13/9 and y is equal to
negative 13/6.
It only has one solution.
So if you think of these as lines, these two lines
intersect in exactly one point.
And a system like this, where it has exactly one solution,
is called a consistent system of equations.
And everything we've been doing so far has been
consistent systems. Let's see if we can stumble upon
something that's maybe a little less consistent.
Let's say we have a system.
Let's say it's 5x minus 4y is equal to 1.
And let's say we have negative 10x plus 8y is equal to
negative 30.
Once again, I'm tempted to do elimination here, because I
have a negative 10x and I have a 5x.
If I take this top equation and I multiply it by 2, I'll
get 10x minus 8y is equal to 2, right?
I just multiplied both sides by 2.
And if we add the left-hand sides, we'll get 0x plus 0y is
equal to negative 28.
So we essentially get 0 is equal to negative 28.
That's crazy.
We know that that's not true.
This can never be true.
We're getting an inconsistent statement.
We're getting a weirdo statement, and that's because
this has no solution.
When you solve a system of equation, it doesn't matter
how you do it, whether it's through substitution or
whether it's through elimination, like I did here.
When you get one of these statements where 0 equals
negative 28 or 5 is equal to 7 or two things that clearly
don't equal each other, when they essentially have to equal
each other in order for the system to work, we call that
an inconsistent system.
And it will have no solution.
So what does it mean for both of these
equations to have no solution?
Let's actually graph these.
I think you'll have a better feel for what it means not to
have a solution.
So the first equation is 5x minus 4y is equal to 1.
Let me put it into slope-intercept form.
So if we subtract 5x from both sides, we get negative 4y is
equal to negative 5x plus 1.
Now if we divide both sides by negative 4, you get y is equal
to negative 5/4x minus 1/4, right?
1 divided by negative 4.
So this is the first equation, right over here, in
slope-intercept form.
Now let me write the second equation in
slope-intercept form.
We have negative 10x plus 8y is equal to negative 30.
Let's add 10x to both sides.
You get 8y is equal to 10x minus 30, and let's divide
both sides by 8.
You'll get y is equal to 10/8, is the same thing as 5/4.
5/4x minus 30/8.
30/8 is the same thing as 15/4.
Oh, and actually I made a mistake here.
When we divide both sides of this equation by negative 4,
negative 5 divided by negative 4 is positive 5/4.
So I shouldn't have had a negative there.
I almost made a blunder.
So there should be no negative there.
Then a 1 divided by negative 4, is negative 1/4.
So going back to the two equations, what do you notice?
Well, when you put them in slope-intercept form, they
both have the exact same slope, 5/4.
but they have different y-intercepts.
So what would their graphs look like?
Let me graph them.
Let's say that that is my y-axis.
That is my x-axis.
This equation, its y-intercept is at 0 negative 1, 4.
Maybe that's that point right there.
And it goes up at 5/4x.
That's a little bit more than one.
It's a 1.25x.
Every time you go to the right 1, you're going up 1.25.
So this line is going to look something like this.
I'm just drawing it rough.
I want you to get the general idea.
That's what that line looks like.
Now this line, its y-intercept is at negative 15/4.
15/4 is what?
3 3/4.
So if this is negative 1/4, its y-intercept is going to be
way down here someplace.
I'll do it in the same color.
Way down here someplace.
Let me continue my x-axis down.
This would be at negative 15/4.
But its slope is the exact same thing.
Every time you go to the right, 1, you're going to go
up by 5/4, so its slope is going to be
the exact same thing.
So what do you notice about these two lines?
They are parallel.
They have the same slope, different y-intercepts, so
they will never intersect.
These two lines will never intersect.
Which means that there is no point on the coordinate plane
on the x-, y-coordinate plane that satisfies
both of these equations.
Remember, this line represents all of the points that
satisfied this equation.
This line represents all of the points that
satisfied that equation.
Notice, no point satisfies both.
There's no point of intersection and that's why
this was an inconsistent system.
Let's do one more.
Let's say I have 4x plus 5y is equal to 0, and I have 3x is
equal to 6y plus 4.5.
Actually, let me do a slightly different one, because I want
to show you all of the different types that we can
see in systems of equations.
Let me clear this.
Let's say my first system is 3x minus 7y is equal to 1.
And let's say my other equation in my system is
negative 6x plus 14y is equal to negative 2.
So, let's try to find the x's and y's that satisfy this
equation here.
And just for a change of pace, let's do some substitution.
Although this is very tempting to do elimination here, let's
do substitution.
You get 3x.
Let's solve it for x.
Actually, let's just do elimination, because this is
just so glaringly prepared for elimination, let's
just do it that way.
So let's multiply this top equation by 2.
And what do we get?
We get 6x minus 14y is equal to 2 right?
I just multiplied every term on both sides by 2.
And now let's add the left sides together.
You get 0 plus 0.
And on the right hand side, you get is equal to 0.
You get 0 is equal to 0, which is always going to be true.
This type of system is called a dependent system.
So remember, when you get a nice clean solution, that's a
consistent system.
When you get something crazy like 0 equals 1, that's an
inconsistent system.
That means the lines are parallel.
When you get 0 equals 0, or 1 is equal to 1 or anything like
that, you're dealing with the dependent system.
Which really means that these are the exact same lines, even
though they might look a little bit different.
And to verify that, let's put them both into
slope-intercept form.
So this top line, you have 3x minus 7y is equal to 1.
Let's subtract 3x from both sides.
You get negative 7y is equal to negative 3x plus 1.
Now let's divide both sides by negative 7.
You get y is equal to positive 3/7x minus 1/7.
That's that first equation.
Now let's put the second equation into
slope-intercept form.
You have negative 6x plus 14y is equal to negative 2.
Let's add 6x to both sides of the equation.
So you get 14y is equal to 6x minus 2.
Then divide both sides of the equation by 14.
You get y is equal to 6/14x minus 2/14.
Well, this is the same thing.
6/14 is the same thing is 3/7x minus 1/7.
Notice, they are really the exact same equation.
So if you want to find x's and y's that satisfy both, let's
think about it.
Let's graph it.
So if that is my coordinate plane, that's the y-axis, that
is the x-axis.
This graph is going to look something like this.
I'm going to draw it very roughly.
It might look something like that, where its slope is 3/7.
So this line is going to look something like that.
That line looks exactly like that.
It's the same exact line.
So when you say, well, what are the x's and y's that
satisfy both of these equations?
Well, it's every x and y that's on these points.
It's that x and y.
That coordinate.
There are an infinite number of solutions.
And when we use the word dependent -- because you can
get to one of these equations from the other.
These equations are dependent on each other.
You can just scale one or the other and rearrange it, and
they equal each other.
So here you have infinite solutions.
Anything that satisfies one line will satisfy the other.
So you just pick an x.
When x is 1, you get 3/7 minus 1/7 that's 2/7.
So 1, 2/7 satisfies both equations.
If you pick x is equal to 0.
0, negative 1/7 satisfies both equations.
And you could pick an infinite number of values for x, solve
for y, and those coordinates will satisfy both equations.
Let me review this a little bit.
So we started off just with the plain vanilla.
When you actually get a solution that is consistent.
These lines actually intersect in one point.
Then you have the situation where you get something crazy,
when you solve your system of equations.
0 is equal to negative 28.
Definitely not true.
This is an inconsistent system.
It has no solution, which means that
these lines are parallel.
They never intersect.
And then finally, if you get something that's always true--
that's just kind of silly how true it is-- this is a
dependent system.
These are going to be the same line.
Then you can verify it by putting both of them into
slope-intercept form.