Hello, I'm Mr. Tarrou. Today we are going to take a look at solving some linear equations.
that you might see in a algebra or in the beginning of you algebra two book.
I've been doing a lot of these lately with one of my classes. And I just wanted to point out to you that
if you take these five steps. You are not gonna be able to solve all equations, especially as these become
your quadratic formulas and stuff like that. Uhh but, they will get you through quite a few of the problems
that you do have in an algebra or beginning of an algebra 2 book. So, lets take a look at a couple of those
And see how these steps help us out.
First of all, we are gonna have 3 minus 4, times lets say x + 2 =7.
This equation has the parenthesis in it. And your not gonna be able to solve for a variable if those
parenthesis in the equation. So, step 1: if you have them get rid of them.
Your gonna do that by taking the number that is right in front of the parenthesis touching it and multiply or distribute it through the parentheses.
It's gonna look like this. Three. Negative four times one is negative four 'x'.
And negative four times nuuhhh positive two is negative eight. Equals seven.
I'm going to keep my work very, very neat, and do one step at a time.
If you're struggling in algebra and you get a lot of mistakes wrong, err uh, questions wrong,
uhh, from small sign errors, this will help a lot. Do one step at a time.
Well. Now it says add like terms. Those terms can have the same variable, or the same number.
If we're talking variables make sure they also have the same exponent.
This equation has a common, er.. like terms of negative eight and positive three.
Those are gonna come together, and the minus sign is for the four not the eight so don't pay attention to that.
The three is positive, the eight is negative. Three minus eight is negative five.
So negative four 'x' minus five equals seven.
We're gonna continue to get this 'x' alone that's on the left hand side.
Everytime you move something now, I'm going to start moving things over the equals sign or under the equals sign.
Anytime you want to take the number over to the other side of the equation you should be thinking
the opposite function. Do the opposite. This 'x' has a five attached to it and we're going to undo that subtraction
with addition. The opposite operation
So, we're going to add five to both sides.
and get negative four x. Equals sign coming straight down. Seven and five is twelve.
Now with the 'x' we have a negative four. It's attached with multiplication.
So, we're going to undo that multiplication with division and get 'x' is equal to negative three. Done.
If you want to check this work, if this is truly the answer, you should be able to I will do it by red chalk
to check the work. Take this answer and plug it back into the original problem. A lot of us forget to do this
or get lazy and they just skip it. But if you want to make sure the negative three is right, plug it back into the
original equation. three minus four times 'x' we just found is negative three.
Negative three plus two parentheses equals seven, WOO! parentheses, you've got to work with those first
inside the parentheses negative three and positive two. The signs are different so subtract those numbers
and keep the sign of the larger number. That's going to be three minus four times negative one equals seven.
Two negatives together, when you multiply make a positive. So now we've got three plus four equals seven and guess what?
It sure is. Now, lets take this check away for a second. Not that we really need to, and focus in
on this equation. Now, if this equation was an inequality. I'm going to change the example a little bit.
If this were an inequality such as less than or greater than or , excuse me, less than or equal to
My mouth doesn't always say what I'm thinking. Whoa, I just made a mistake there
Because I was about to tell you that when you solve an inequality, you follow the exact same steps as
you do in an equation and you do, but I got a little mistake in here somewhere. Where is it?
When you divide by a negative number, this is the only difference when you deal with inequalities
compared to equations. When you divide by a negative number you've got to remember to come back
up here and change the direction of that inequality. When I divide by that negative 4 and got negative 3
if it were an inequality and now it is, you have to change the direction of that sign.
That is the only difference between solving inequalities and solving equations.
Let's take a look at a mistake that I see very, very often with students that are just starting off in algebra.
I'm going to keep this equation almost the same, but I'm going to take the four out and replace
it with just a straight minus sign. Now in that case a lot of students want to take this three and multiply it
through the parentheses. Now I'm not going to highlight that because that's not right!
See, before there was a four there touching, right against that left parentheses. Touching means multiplication.
That says the three is subtracted to whatever is in the parentheses.
So, what is in there? What are we supposed to do with the negative sign?
Well, there is a negative one there that doesn't have to be represented. We can just have the minus sign
but there really is like a multiplication of negative one
and then negative one is what's going through the parentheses giving you this
three. Negative one times one is negative one x. Negative one times positive two is negative two
equals seven. Again with the like terms. Three minus two is one so negative 'x' plus one equals seven
Bring the one over. Whenever you move something over the equal sign you want to do that
with the opposite function and the opposite of addition was subtraction. We still have a minus
in front of that 'x'. That means it is not solved for yet. You get that negative one away by dividing
and when you're all done if you thought it was a hard question, BAM! celebrate.
Okay, so now lets take a look at something else. Lets take a look at 2/3x + 1/5x = 10+ 2x
You're like okay whoa I am no good at fractions. First of all, you do need to be good but I'm going to show you
just for now if you have an equation with fractions and you do not want those there you can get rid of them.
A fraction bar means division and if you don't have or if you do have a division that you do not want
you can undo that division by multiplying. Do the opposite of whatever that math function is.
Well, so I'm telling you that I'm going to multiply both sides of the equation by a number
Now I can't just pick a number. I have to multiply by something to cancel out the three
and I have to multiply by something that will cancel out the five.
I want to get rid of both of these. So, five, three doesn't go into five. ten [tap tap tap]
Multiples of five. ten, three doesn't go into ten. Fifteen, yeah three goes into fifteen
So, I'm going to take fifteen, multiply both sides of the equation by this value
I'm going to show the work in case you're not very strong with fractions.
and we will cancel them later. 15 times 2 is 30 over 3x plus 15 times 1 is 15 over 5x equals 15 times 10
is going to be 115 (really 150). So hopefully I can keep doing this in my head. Plus 15 times 2 is 30x
I still have fractions, but not really. Because i multiplied by 15 this will cancel out.
30 divided by three is 10 and 15 divided by 5 is equal to 3. Now, all my fractions are gone
and I just have whole number answers. So, I'm going to add like terms because if I do have like terms
on the same side of the equal sign. Equal signs break an equation up into two parts. We have 10 and 3 is 13.
I'm just copying everything else down. I kind of moved my equals sign over. I don't like the fact that I did that.
So we have once again 13x equals 115 plus 30x. Now, when you solve an equation and you have a variable
on both sides of the equation you can put that 'x' anywhere you want, left or right, but most text books
keep the variable to the left so, you don't have to but you might just want to always take the x that's on the
right and I'm going to move it over with subtraction. Why am I doing that?
Because it's already positive. I want to undo that. That's going to cancel that 30 out. 13 minus 30 is 27, -27.
Let's try -17x equals 115 and I'm going to divide both sides by -17 because it's still with the 'x'
and it's touching so I'm going to divide both side by -17 and x= -115/17 and I'm sure if you made that a decimal
that would probably be okay. I think I'm out of time. Let me go check.
[silence]
Hello, I've got about five minutes left. Let's take a look at this. This looks quite a bit different than my previous
two examples and I've got a fraction on each side. If you have an equation with only one fraction on each side
you can solve that equation with cross multiplication. Well, okay, but what if you don't have just one fraction on each side.
That looks a lot more complicated than the previous question I just had up here.
Well, I can still look at these fraction bars like division because, well, they are.
And I can still, like my last example, multiply both sides of my equation by something both of those numbers out
So what number does 3 and 4 both go into? 4, 8, oh 12. okay. So lets multiply the entire thing through by 12
and see what happens. Now this is a little bit more complicated than the other one. This left side is a big old fraction bar.
Now there's two terms up here, but this fraction bar is holding those two terms together. It's one term.
It's a grouping symbol, actually. So when I multiply by 12 I only have to do it once.
So, this is going to be 12 times 'x' plus 2 over 3 and see, we're going to have a cancellation here in a minute.
equals, same thing over here. One term and another term even though there's a lot going on here
this is one term so 12(4x-1) over 4 plus 2 times 12 is 24. I still have my fractions.
I'm going to erase because, you know, I have limited space but you might want to keep writing down on your paper
What's 12 divided by 3? 4. So, take that out. Make it four. I don't need this fraction bar anymore.
12 divided by 4 is 3. So before I put the 12 through the parentheses I'm able to cancel it with the bottom
because it's in front of a parentheses like that. It's a factor. And then of course we still have the 24.
This may look like more of your difficult questions at the end of an Algebra I book when you are first solving
multiple step equations. Lets just take it one step at a time. You've got parentheses. Step one. Get rid of them.
So 4 times 1x is 4x. 4 times 2 is 8 equals...again with the distributive property because there is a number in front of that parentheses.
If there were division or an exponent it would be a little more complicated but there is not so 3 times 4 is 12
3 times -1 is -3 and 24. Okay. If you have like terms, put those together. Again my like terms don't have variables.
I probably should have done them like that but we have a positive 3 and 24. Those will come together and make 21.
So, 4x + 8 = 12x +21 and again I have a variable on both sides. Now, if you want to avoid negatives
take these small 'x' and move it underneath the larger one and just make your movements so that you have a
positive answer. If you're a student that really wants those 'x's on the left because you're more comfortable with it
well then subtract both sides by 12. I'm going to move the 4 though this time, so I'm going to subtract 4 over
from the left and right hand side. I'm going to erase it though because I'm starting to run out of space,
but I am going to move the 4 over with subtraction so Im going to get 8x equals... excuse me that's an 8.
8 equals 12 minus 4 is 8x plus 21. Hows that 21 go away from the x? How do I make that 21 go to the left hand side?
when its already positive? You subtract it. So I'm going to subtract both sides by 21.
And I guess Im going to have to make some more room here anyway so I'm going to bring this over and get
what is that, -12, 13? 13. So, -13 equals 8X and the 'x' still has an 8 with it
So, I'm going to move that 8 away by doing the opposite of the math operation already here
which is multiplication so I'm going to divide both sides by 8 and get -13 over 8 is my answer for 'x'.
Now remember I'm going to be out of time, I can't show this but if you want to make sure this actually works
you've got to plug it back into the original problem and see if both sides match. Thank you very much!