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>> Math is cool and you can do it.
Okay let's say that you were given an equation and someone asked you
if a particular number was the solution to that equation.
The question is how do you decide whether or not that number is a solution to an equation?
Okay, here are the steps first, in the equation everywhere there is the variable replace it
with a number you're given.
Second, you're going to simplify each side
of the equation separately using the order of operations.
Third, if both sides simplify to the same number then that number is a solution
because then you'd have both sides equal.
So first of all you have to be given an equation,
note if it is a solution it might not be the only solution.
So another question, is two a solution to 3x plus 9 equals 2x plus 11?
Now the way I like to do this is I underline the entire problem and then
under the equal sign I'm going to draw a bar so it kind of looks
like a big T the reason I do this is to make sure that I'm really just paying attention
to one side of the equation of the time, at a time.
So on the left hand side I'm going to replace the x with a 2, so 3x means 3 times 2 plus 9
and then I'm going to use the order of operations so that's just 6 plus 9 which is 15.
Now on the right hand side I'm also going to plug in 2 for x and use the order of operations,
so I have 4 plus 11, which is also 15.
So when I replaced a 2 I got the same number on both sides
when I simplified it, so yes 2 is the solution.
Here is another one.
Is 6 the solution to 2x plus 7 equals 3x?
Alright let's do the same thing here, I'm going to underline the entire problem
and under the equal sign I'm going to put the bar and we're going to replace the variable
by x. Alright, we're going to plug that in for x, so we have 2 times 6 plus 7
and then we're going to see what this side simplifies
to using the order of operation, 12, plus 7 is 19.
And now let's do the other side.
We're going to plug in 6 for x over here and that gives me 18.
They're both simplified to one number, no not a solution, so no 6 is not a solution.
Alright let's look at this one.
Is 5 a solution of x minus 3 equals x minus 3?
Alright, let's underline it, the bar.
Alright, so we're going to replace the x with a 5, so I have 5 minus 3 is 2 and then we're going
to put 5 over here, 5 minus 3 is 2.
So yes, yes 5 is a solution.
Okay, now what if I had taken the same problem and asked if 10 was a solution?
So we know 5 is a solution right?
So let's go back and see if 10's a solution.
Is 10 a solution?
Hum, well let's put in 10 for x so I get 7 on the left hand side and I'm going to put in 10
for x over here and hum, guess what 7's a solution too.
So remember when I said that sometimes it's not the only solution?
Here 7's a solution, 5's a solution, and in fact,
any number you put is going to be a solution here.
X minus 3 equals x minus 3 for any value of x
because the 2 sides are exactly the same, so you might notice that.
So actually the solutions, later you'll get into this will be saying
that all real numbers are solutions, everything.
Alright, let's try this one.
Is 2/3 a solution if 18x equals 12?
Well let's do our thing here put in the T and we're going to plug in 2/3 for x. I'm going
to write 18, now something in multiply 18x means 18 times x since I'm going to multiply
with a fraction I'm going to write this as 18 over 1 and for x I'm going to put in 2/3 okay.
And let's see if anything cancels here because we have to simplify it.
6, so I have 6 times 2 over 1, well 12 over 1 is just 12 you could first write 12 over1.
Now on this side notice that there is no variable at all
so there's nothing to plug in and your answers 12.
So you got the same thing on both sides didn't you?
When you plugged in, plug in 2/3 the equation is true so yes.
2/3 is a solution.
Here's another one.
Is 13 a solution to x minus 6 equals x?
Well let's see.
Underline it and let's plug in 13 for x. So 13 minus 6 is 7 and here we put in 13
for x and these are not the same.
So when I simplified the left hand side I got a 7
but when simplified the right hand side I got a 13.
So nope, 13 is not a solution.
And if you look at that equation x minus 6 equals x won't have any solutions.
You're not being asked that but, because that's like saying if you take a number
and subtract 6, you have the original number.
That doesn't make any sense.
That's like saying, you know however much money you have
if you take away 6 dollars you still have the original amount of money.
Hum doesn't make any since.
Let's try one more where you have a exponent in it.
Is 2 a solution of x squared plus 5x equals 14?
Alright, so let's do our usual underlining and we're going to put in 2 for x and notice 2 is
in two places on the left hand side, so we have to put it in for each value of x
because we're trying to find out if 2's a solution.
And doing my order of operations 2 squared is 4 plus 10 I can do those at the same time
because I'm going to add last and that's 14.
Now the left side there is no x at all, so I could write the 14 down here or leave it
up there, sometimes I do that right next to each other, I see what I got on both sides.
Yes, I do get the same number.
So the answer to this question is yes 2 is a solution.
Alright, let's work on this one.
Is 5 a solution of 2x squared minus 20 equal 6x?
Alright now, remember 2x squared does not mean 2x times 2x.
The base is x, so only the x gets squared, so it's very important you're careful here.
So before plugging in the x I'm going to write out 2x squared is 2 times x times x,
don't have to do that but that's one possibility.
Now I'm going to put in 5 for each value
of x. Let's see I have 10 times 5 that's 50 minus 20 which is 30.
Alright, now on the other side we're going to plug in 5 for x, 6 times 5 which is 30
and yep we got the same number on both sides.
So yes, 5 is a solution.
Here's another one.
Is 3 a solution to x squared equals 5x?
Try this one on your own.
Okay, so we'll underline it as usual make the T and put in 3 for x.
So we could either write x times x first and then write 3 times 3
or we could just put in 3 squared.
And 3 squared you can do that in your head probably is 9, I mean 3 times 3 is 9
and then we're going to put in 3 for x on the right hand side.
5 times 3 is 15 and so no, no 3 is not a solution.
So here's the cool thing about algebra, is that if you are solving equations,
which we'll be doing and you think you got a solution what you could do is go back
and check to make sure you're right.
This is exactly how you would check your work you would go back and plug in that value
for the variable and make sure you really do get a true equation, which means you're going
to get the same number on both sides of the equation the left side
and the right side after you plug it in.
So that's what's really nice about algebra, if it doesn't work then you know you made a mistake
and you can go back and do the problem over again.
In arithmetic it's not like that, sometimes you make a careless mistake and there's no way
of checking except just doing it over and over again
and sometimes you make the same mistake over and over again.
Algebra gives you the luxury of knowing whether you're right before you turn in your test.
I hope you learned something, remember math is cool.