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This is going to be the first of a series of videos about equations with
two variables, tables of values
and graphs.
So let's get started.
Let's say I've got the equation x plus y equals 5.
Well this is an equation with two variables -
the x is a variable and the y is a variable -
and if somebody asked me
what y is,
I couldn't answer that question
because that would depend on what the x is.
If they told me what the x was, I could then answer it.
Let's say that said that x was 2.
Well if x is 2,
then I'm thinking 2 plus some number equals 5
and that number would be 3.
And I could think that formally - in other words, algebraically - I could write x plus y
equals 5,
and I could put a 2 in instead of the x.
So 2 plus y
equals 5,and then of course I'd subtract the 2 from both sides
and I'd get y equals 3.
So I'm gonna make a
table - this is called a table of values -
and over here I'll write x and I'll write y
and this way I'll know that
when x is 2
y is 3.
Now, what if they told me that x was 1?
Okay, I would go back and do the same thing again
and
I would figure that one...
that if x was 1,
y had to be 4.
And I could check that. I could think that
1 plus 4 is 5,
so this works.
And I could keep going like this.
If they said
x is zero,
I'd have a 5 here.
If they told me x was negative 1,
then
once again I would plug in
a negative 1 over here to try to figure out what my answer is -
let's go over here and figure that out...
So if x is negative 1, I've got negative 1 plus y
equals 5,
which means I'm gonna add 1 to both sides
and I'll get y equals
6.
And I could do one more.
If they told me x was negative 2
then I'm gong to have a 7 over here. Now if you think about it,
I could keep writing different values for x
and for each different value
I'm gonna have some number that matches it for y.
So what that means that is,
first of all, that each of these pair of numbers
is a possible solution to this equation.
I've got to take each pair as a pair -
so 2 plus 3 equals 5,
1 plus 4 equals 5, 0 plus 5 equals 5,
and so on. There would be infinite number of possible pairs.
It's not like I can say there's just one solution.
Each one of these
pairs of numbers
is a solution.
Now one of the things you'll find mathematicians saying, your teachers
and your book and so on, is that these numbers, this pair of numbers
satisfies the equation x plus y equals 5.
And here's all that means.
If I take the equation x plus y
equals 5
and I take... let's take this last pair of numbers, negative 2 and 7
and I plugged those numbers in
in the original equation... so instead of x I'm gonna write negative 2 and
instead of y I'm going to write 7,
and then I take that and do the arithmetic here... negative 2 plus 7 is 5,
5 equals 5... so any pair of numbers
that makes the equation balance
is going to satisfy the equation.
Let's take a look at another one.
Here we have a somewhat more complex equation
and what I'm going to do
rather than try to plug numbers into it as it is, is make it into a form that'll
make more sense to me.
So this is called
solving for y.
And what that means is I want to get the y by itself, I want to isolate it.
so what I'm going to do is I'm going to subtract 4x
from both sides of the equation,
and that means I'll just have negative 2y
on the left.
On the right side I'll have negative 4x
plus 6.
And then what I want to do is divide by negative 2,
so the y will be
isolated.
So let's divide by negative 2.
well that means I have to divide everything by negative 2.
So dividing the left side, I'm just going to get y.
Dividing the right side, I divide negative 2 into negative 4x,
and that will give me 2x
and then I divide
negative 2 into 6 and I get negative 3.
and this will be a lot easier for me to work with.
When somebody gives me an x
I'll be able to plug that in right here
and immediately know what the y is.
So let's do another table of values and see how this works.
Okay, so here's my x column
and my other column is a y column, but if you think about it, since y equals
2x minus 3,
I could also call this a 2x minus 3 column.
Now you don't have to write this over here,
but it might make things easier, because that way when somebody gives me a number -
let's say somebody says
x is negative 2,
then all I have to do is look over here
instead of going back to my original equation.
Just go over to here and say... okay negative 2,
2 times negative 2 is negative 4,
negative 4 minus 3
is negative 7.
Let's plug in a few more numbers.
If I've got a negative 1,
I'm going to have 2 times negative 1 - since negative 1
is an x
and I have this x over here -
2 times negative 1 is negative 2,
negative 2 minus 3
is negative 5.
If
x is zero,
then
I'm gonna plug a zero in here. So 2 times zero is zero
minus minus 3 is negative 3.
Let's just do a couple more...
If x is 1, I've got 2 times 1 is 2
minus 3
is negative 1.
And if x is 2,
2 times 2 is 4,
4 minus 3 is 1.
So each of these pairs of numbers, once again,
is
a solution, as a pair,
a solution
to the original equation.
I can go back to the very first equation I had, to this one,
and plug those numbers in
and since I could do that,
I would be able to say those numbers satisfied the equation. Let's check and
just make sure. So let's see if the numbers negative 2 and negative 7
satisfy the original equation.
So let's write the original equation again.
4x
minus 2y
equals 6.
And now I'm going to say
x equals negative 2
and y equals negative 7,
that's my pair.
So let's plug the numbers in.
So 4 times
negative 2
minus
2 times negative 7
equals 6.
4 times negative 2 is negative 8,
negative 2 times negative 7... negative times negative is positive, so that's
plus 14
equals 6.
negative 8
plus 14 do equal 6.
So
this
over here,
x equals negative 2, y equals negative 7, is a solution
to the original equation.
Or we can say it satisfies the equation.
Again,
each one of these pairs is a solution, each one of these pairs if
you plug them in
will satisfy the equation
and there is
an infinite number of pairs
that will satisfy the equation.
Okay? So
I'm gonna stop this now and
I'll pick up again with another video where we go and start talking about
graphs.
Okay, see you soon.