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>> Narrator: We're continuing to work with linear equations and two variables.
And in this section, we're going to start off finding ordered pair solutions
to a linear equation if I give you a value of either the X coordinate or the Y coordinate.
So here's an example, find the ordered pair solution
of Y equals 3 halves X minus 4 corresponding to X equals 6.
What this is saying is you want the ordered pair where you're going to put in 6 for X.
So what you're looking for is the Y coordinate.
So how would I get that?
Well, I've got Y equals 3 halves, X minus 4.
I'm going to plug-in 6 for X. All right.
Now, I'm going to write 6 as 6 over 1.
You don't have to do that.
And then, we simplify the right hand side.
So let's see, 2 goes into 6 three times, so this gives me 9 minus 4.
Everybody agree with that?
Because 9 over 1 is 9.
So we have 9 minus 4 is 5.
So the ordered pair solution that we're looking for is (6, 5).
And that's our answer.
Let's try it corresponding to some other number.
All right.
This time it says find the ordered pair solution
of Y equals 3 halves X minus 4 corresponding to X equals negative 4.
So again, we're given the X coordinate of the ordered pair.
So for the X coordinate we're going to plug-in a negative 4 and we're trying to find the Y value.
So we're going to plug-in what for the X value?
Negative 4.
So again, I'm going to write that as negative 4 over 1 since I'm multiplying by a fraction.
And then, we simplify this.
That's going to be a negative right because of the positive times a negative here.
And the 2 goes into 4 two times, so I have 3 times negative 2 so that's negative 6.
Be careful with your signs.
And negative 6 minus 4 is the same as negative 6 plus negative 4 that's negative 10.
So the ordered pair corresponding to an X is 4 is this ordered pair, (negative 4, negative 10).
All right.
Here's another 1.
Find the ordered pair solution of 2X minus 3Y equals 12 corresponding to X equals 3.
All right.
So again, I know the X coordinate is going to be 3 so I could put that in.
Now we got to take this equation over here 2X minus 3Y equals 12 and we're going to plug-in 3
in place of X. And now, I don't just get to simplify the right hand side of the equation;
I'm going to have to solve for Y. So let's see.
First, I'm going to simplify the left side, 6 minus 3Y equals 12.
Then I'm going to subtract 6 from both sides to get negative 3Y equals 6.
Then I'm going to divide by negative 3 and I finally get the Y value of negative 2.
So when it's not solved for Y like in that previous one, there's a lot more work
when you're plugging-in that value for X. Now, you should make sure
that you didn't make a mistake by checking to make sure that (3,
negative 2) really is a solution to that.
So if I'm going to take 2X minus 3Y equals 12, a quick check put in 3 for X,
negative 2 for Y and make sure it's true.
So that's 6, this is minus negative 6 or plus 6 equals 12 so you just want
to do a little double check there.
So therefore, (3, negative 2) is the ordered pair we're looking for that problem.
All right.
How about if we're asked to fill in the blanks
to find the ordered pair solutions to Y equals 2X minus 5.
So we're trying to find three ordered pairs.
I could see the X value in two of the ordered pairs and the Y value in one.
So we need to do three things.
We need to find each ordered pair.
So let's see.
Let's work on the first one.
If X is 0 and we put that in this ordered pair, what will Y equal?
We'll if I'm going to plug-in 0; that gives me 2 times 0 minus 5, which is negative 5.
So when I plugged in 0, Y was negative 5.
All right.
Second one.
It tells me what the Y value is so I have to take this equation and I have
to plug-in 5 for Y to figure out what X is.
So 5 equals 2X minus 5, now I've got to solve for X so I have to add 5 to both sides.
So basically, I have 2X is 10.
Hopefully, you can see that X is also going to be 5.
All right.
Last one, we're going to plug-in negative 1 for X. So we have 2 times negative 1 minus 5.
Y would be negative 7.
All right.
So what we found are three ordered pair solutions, we filled in the blanks.
If you want to write these as ordered pairs, remember how they look, (0, negative 5), (5,
5), and (negative 1, negative 7).
If you were to graph those three ordered pairs on the XY-plane,
the rectangular coordinate system and draw the line through those three points,
you would have all the ordered pair solutions on that line.
So let's go ahead and do that.
All right.
So if we look at these ordered pairs: (0, negative 5), (5, 5), (negative 1, negative 7).
Notice I have plotted those three points on this graph
and now we simply draw the line through those three points.
So once you draw the line through those three points, you can convince yourself
that every point on that line is a solution.
So what you could do is a little check remember is figure out one
of the points that line is going through.
So for instance, I see this point right here.
That looks like (3, 1).
Let's make sure (3, 1) is a solution.
Remember, so we're going to put in 3 for X, right.
You're going to put the 3 in for the X and we're going put in the 1 for the Y
so that will give me 1 equals 6 minus 5.
Yep. So looks like I got the correct graph.
So we've done a few things here.
We found -- we filled in this chart to get three ordered pairs on that line
and then we also went ahead and graphed the three ordered pairs and drew the line
between them or through them and that gives us all the ordered pairs
that are solutions to that equation.
Now, here's the thing you may wonder.
How did, you know, I come up with, well, I'm going to plug-in 0 for X, or 5 for Y,
or negative 1 for X. When you have a linear equation, you can choose any number you want for X,
plug it in and you'll be able to solve for Y. That'll give you an ordered pair.
So there are no restrictions on what you put in for one of the variables.
So you pick a variable, you pick a number for a variable,
and then that's half of your ordered pair.
So in general, there are no restrictions but there's ways
of picking easy numbers to start with.
So here's an example, find three ordered pairs for Y equal 2X minus 3.
We're just going to make a table, and I'm going to pick anything I want for X
and then I'm going figure out what Y is based on that X. So let's do that.
Any number you want for X. All right.
Let's just pick 0, 1, and 2.
Simple enough?
So let's put in 0 for X, and what does that give me for my Y values?
So if I simplify that that gives you 0 minus 3, Y would be negative 3.
What order pair is that?
(0, negative 3).
Let's do the next one.
We're going to put in 1 for X, so Y would be 2 minus 3, or negative 1.
What ordered pair is this?
Here's your X, here's your Y, (1, negative 1).
All right.
Now, let's put in 2 for X. So I have 4 minus 3 is 1,
and I've just found three ordered pairs for my line.
And I'm going to another video to show how to get equations in an easy form to do this
and also how to pick easy values for X.