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>> This is part 2 of solving puzzles and riddles,
otherwise known as word problems, using two variables.
So we'll be doing a couple more examples.
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>> Next problem.
A first number plus twice a second number is 23.
Twice the first number plus the second number is 31.
Find the numbers.
All right, we're looking for two numbers,
which we're calling a first number and a second number.
So that's why I'm defining them.
First number, let's call that X.
And the second number let's call it Y. Now we need
to write two equations.
The first sentence gives us the first equation.
A first number, or X, slash, twice a second number,
so that would be 2Y, since Y's the second number, is 23.
First equation.
Second equation is twice the first number, all right,
so T times X plus the second number,
which is Y, is equal to 31.
All right, so those are our two equations, and now we have
to decide if we're going to use the substitution
or the addition method.
I think I'm going to use the substitution method.
And I'm gonna move this second equation down here,
so that I could easily solve for X in the top equation.
So if I solve X I need to subtract 2Y from both sides,
so that X equals 23 minus 2Y.
Okay, so that's what X is, right?
Could put a little star so I remember what I substituted.
And I need to put that over here in the other equation.
So I need to put that in for X into the 2X plus Y equals 31.
So we have 2 times 23 minus 2Y plus Y equals 31.
All right, now we have an equation with only one variable.
So we could do the distributive property
and simplify the left hand side to begin with.
Remember to distribute the two.
46 minus 4Y plus Y equals 31.
And then we're gonna simply the left hand side,
-3Y plus 46 equal 31.
And at this point what I'm gonna do is I'm gonna add 3Y
to both sides, so it's on the positive side of the equation,
on the right hand side making it positive 3Y.
And since the Y values are gonna be on that side,
I'm gonna subtract 31 from both sides.
Visually that makes it easier for me, but you do not have
to solve it exactly the same way I do.
So let's see, this gives me 15 equals 3Y,
and then we'll just divided both sides by 3.
And that'll give me 5 equals Y or Y equals 5.
All right now, I don't even remember what Y stood for,
remember it's a word problem or a puzzle.
Hum. Y was the second number.
So the second number's 5.
We need to find the first number.
Remember where we circled what X was equal to?
Now that I know what Y is, right, I know what Y is,
I'm gonna plug that in for Y up here.
So I have X equals 23 minus 2 times 5.
Putting in 5 for Y. 23 minus 10 equals 13.
And so now I also know the X value.
Now that you know that the numbers are 13 and 5,
you need to make sure that you check
that in the original words given,
and then write your answer in a sentence.
So let's do that.
It says, a first number plus twice the second number.
Well the first number's 13, right?
And then it says, plus twice the second number.
Well the second number was 5.
So twice that is 10.
And that gives me 23, which is exactly what it's supposed
to do.
So I'm checking in words what it says.
And it also says twice the first number.
Okay, the first number's 13, so twice that is 26,
plus the second number, which is 5, well let's add up to 31.
And yes, it was supposed to add up to 31.
So these must be the correct numbers.
So now you write the numbers are 13 and 5.
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>> All right, here's the next problem.
The sum of two numbers is 16.
The first number, plus 2 more
than 3 times the second equals 18.
Find the numbers.
Again, we have a situation where we're looking
for a first and a second number.
Why don't you put this on pause and try
to write the two equations and then come back and see
if you get the same thing I get.
All right, so we have X and Y for the two numbers,
the sum is 16, that's the easy part.
So X plus Y equals 16.
That's one equation.
The second one, the first number,
all right so we're gonna start with the first number,
plus 2 more than 3 times the second.
Ooh boy. So we're gonna add something in here.
We're going to add 2 more.
That means we're gonna add 2 to something.
To what? 3 times the second.
But the second is Y, right?
So that's 3Y.
So there it is, X the first one, slash,
2 more than 3 times the second, equals 18.
Wow, those are our two equations.
Now at this point you can use the substitution
or the addition method.
I'm gonna take the second equation and simplify it,
so I've got the variables on one side and constants on the other,
so to do that I need to subtract 2 from both sides.
So the second equation becomes X plus 3Y, if I subtract 2
from both sides I get 16.
So I've got really these two equations right here.
Okay? All right, now it's up to you which method you'd
like to use to solve for X and Y. How
about we multiply the bottom equation by -1,
so that when I add the X's they're gonna cancel.
All right?
So I'm gonna show it over here.
The first equation is the same.
And I'm gonna take the second equation
and multiply both sides, [inaudible] both sides by -1,
so that gives me a -X, a -Y, and a -16.
And if I add up I get 2Y is 0.
Remember, that is something, zero's a number,
so when we divide by 2 I've got y equals 0.
So it looks like the second number is zero.
Now let's go back to one of the equations
and figure out the other one.
Well the top equation was X plus Y equals 16.
So if Y is 0, X has got to be 16.
So it looks like my numbers are 16 and 0.
All right, well let's check.
The sum of 2 numbers is 16, yep,
because 0 plus 16 certainly does add up to 16.
Okay, now it says the first number, 16 right,
plus 2 more than 3 times the second.
All right.
Well what's 3 times the second number?
Well 3 times 0 is 0.
Then if I add 2 to 0 I get 2.
Okay, so I just get 2 and I add that up I get 18.
So it does check.
So the numbers I was looking for was 16 and 0,
and that's how we finish.
The numbers are 16 and 0.