Old school equations with Sal
Order of Operations examples Evaluating expressions using order of operations
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- Let's do some order of operations problems, and for
- the sake of time I'll do every other problem.
- So let's start with 1b.
- 1b right there.
- They have 2 plus 7 times 11 minus 12 divided by 3.
- So just remember, the top priority is always going to be
- your parentheses.
- So you have your parentheses-- Let me write it this way.
- Your top priority's going to be your parentheses, after
- that you're going to have your exponents, after that you have
- multiplying and dividing, and after that you have addition
- and subtraction.
- So let's remember that and tackle these order of
- operations problems.
- So priority, there's no parentheses here, there's no
- exponents, so the priority's going to go to multiplication
- and division.
- So you could view this as being equivalent to-- So we're
- going to do our multiplication before we do any addition or
- subtraction, and we're going to do our division before
- doing any addition or subtraction.
- Problem 1b is exactly equivalent to this, the
- parentheses are just-- I'm reinforcing the notion that
- I'm going to do my multiplication and division
- before I do the addition and the subtraction.
- So 7 times 11 is 77, and then 12 divided by 3 is 4.
- And the rest of the problem was 2 plus this thing, which
- is 77, minus this thing.
- And here, since everything is in addition or subtraction,
- let's just go left to right.
- 2 plus 77 is 79 minus 4, which is equal to 75.
- So 1b is equal to 75.
- Let's do 1d.
- This is a nice hairy problem right there.
- So 1d.
- It says 2 times 3 plus 2 minus 1.
- Closing two parentheses, all of that over 4 minus 6 plus 2
- minus 3 minus 5.
- Let's see if we can simplify this a little bit.
- As we said, parentheses take our priority.
- So let's do the parentheses first. 2 minus 1.
- 2 minus 1 is just 1.
- 3 minus 5.
- That is minus 2, or negative 2, I should say.
- 6 plus 2 is 8.
- Now let's keep looking at the parentheses to see where we
- can simplify things.
- We have this parentheses right here.
- So 3 plus this 1 is now going to be equal to 4.
- Actually, let me rewrite it.
- So we're going to have 2 times this whole expression, 3 plus
- 1, so it's 2 times 4.
- That right there is 4.
- All of that over 4 minus 8, that's negative 4.
- This right here is negative 4.
- And then minus this negative 2.
- So minus negative 2.
- 2 times 4 is 8, so this whole thing simplifies to-- A minus
- of a negative, that's just the plus of the plus, the
- negatives cancel out.
- So this whole thing simplifies to 8 divided by negative 4 is
- negative 2 plus 2.
- So it equals 0.
- So this big, hairy thing simplified to 0.
- Now let's do 2b.
- Let me clear some space here.
- I'll leave the order of operations stuff there.
- Let me clear that and let me clear this.
- All right, 2b.
- Evaluate the following
- expressions involving variables.
- Fair enough.
- So they wrote 2y squared, and they're saying that x is equal
- to 1, which is irrelevant because there is no x here,
- and y is equal to 5.
- So if y is equal to 5, this thing becomes the same thing
- as 2 times 5 squared.
- And notice, I put parentheses there.
- I could have written this as, this is the same thing as 2
- times 5 squared.
- And if you look at the order of operations, exponents take
- priority over multiplication.
- That's why in my head I just automatically put those
- parentheses.
- We're going to do the exponent first.
- So this is 25, so you get 2 times 25 is equal to 50.
- So that is 2b, this is equal to-- use a darker color --that
- is equal to 50.
- Let's do 2d.
- They're giving us y squared minus x, whole thing squared.
- x is equal to 2 and y is equal to 1.
- Well, we just substitute.
- Where we see a y we put a 1.
- So this is going to be 1 squared minus x squared--
- Sorry, minus x, not x squared.
- So we just put a regular x there.
- That's where we put a 2.
- And then all of that squared.
- Well 1 squared is just 1, so that is just 1.
- 1 minus 2 is negative 1.
- And then we're going to want to square our negative 1, so
- that will be equal to positive 1.
- So that is equal to 1.
- Negative times a negative is a positive.
- All right, let's do 3b.
- Doing every other problem.
- I'll do it in yellow.
- Evaluate the following
- expressions involving variables.
- All right.
- Same idea.
- So they're giving us 4x over 9x squared.
- Oh, actually I said I'd do 3b, I was doing 3a.
- So here we go.
- We have z squared over x plus y plus x squared
- over x minus y.
- And they're telling us that x is equal to 1, y is equal to
- negative 2, and z is equal to 4.
- So let's just do our substitutions first.
- So z squared, that's the same thing as-- I'll do it in a
- different color --4 squared over x, 1, plus y, negative 2,
- plus x squared, that's 1 squared, over x,
- which is 1, minus y.
- y is negative 2.
- So this is going to be equal to 4 squared is 16 over 1 plus
- negative 2, that's 1 minus 2-- it's just a negative 1 --plus
- 1 squared, which is 1, over 1 minus negative 2.
- That's the same thing as 1 plus 2.
- So it's 1/3.
- And so this will be 16 divided by negative 1.
- We could write that as that's equal to negative 16 plus 1/3.
- Now if we want to actually add these as fractions we could
- have a common denominator.
- Negative 16 is the same thing as minus 48 over 3, or
- negative 48 over 3.
- If you take 48 divided by 3 you'll get 16, and I'm just
- keeping the negative sign.
- And then you add that plus 1/3.
- We have a common denominator now, 3.
- Negative 48 plus 1 is negative 47.
- So our answer is negative 47 over 3.
- Problem 3d.
- Same type of situation.
- x squared minus z squared over xz minus 2x times z minus x.
- x is equal to negative 1, z is equal to 3.
- Let's do our substitutions.
- So this is x squared.
- That's minus 1 squared.
- Minus z squared, so minus 3 squared.
- All of that over x times z.
- x times z is minus 1 times 3, minus 2 times x, x is negative
- 1, times z minus x, times 3 minus x.
- x is negative 1 minus x.
- Wherever we saw an x we put a minus 1.
- So this is going to be equal to-- Remember, you do your
- exponents first. Well,
- parentheses first, then exponents.
- So we have negative 1 squared, that's just a positive 1.
- 3 squared, that's just a positive 9.
- So our numerator becomes 1 minus 9, that's minus 8 or
- negative 8.
- And then our denominator.
- Negative 1 times 3 is negative 3.
- And then let's go to our parentheses here.
- We have 3 minus negative 1, that's the same thing as 3
- plus plus 1.
- So that right there becomes 4.
- So our denominator becomes negative 3 minus 2 times
- negative 1 times 4, so that's negative 8.
- Minus negative 8.
- Minus of a negative is the same thing as a plus.
- So this whole thing becomes negative 8 over negative 3
- plus 8 is 5.
- So it's negative 8/5, minus 8 over 5.
- All right, let me clear up some space just so we can
- reference this problem properly.
- Let me clear all of this out of the way.
- These are interesting now.
- Problem 4: insert parentheses in each expression to make it
- a true equation.
- Fascinating.
- So 4b.
- You have 12 divided by 4 plus 10 minus 3 times 3 plus 7 is
- equal to 11.
- So let's see what happens if we just do traditional order
- of operations, and I'll do a little bit in my head because
- this is going to take some experimentation.
- Oh yeah, this is 4b, 12 divided by 4--
- Yep, that's the problem.
- So if did 12 divided by 4 first, and we would get 3.
- So let me just do this in yellow.
- So if we did regular order of operations this would be a 3.
- This right here would it be a 9.
- So you would have 3 plus 10, which is 13, minus 9, 13 minus
- 9 is 4 plus 7.
- Actually, that seems right.
- Let me make sure I did that right.
- 3 plus 10-- Right, that looks right.
- So we really just have to do regular order of operations.
- So it already looks like a true equation.
- So if you do 12 divided by 4 plus 10 minus 3 times 3 plus
- 7, I think it turns out right.
- Let me confirm.
- Make sure I'm not making a mistake.
- 12 divided by 4 is 3 plus 10 minus 3 times 3 is 9 plus 7.
- This is equal to 13 minus 9, which is equal to-- So all of
- this is equal to 13 minus 9 is equal to 4 plus 7 is, indeed,
- equal to 11.
- So that one wasn't too bad.
- You actually wouldn't have to put any parentheses to make
- this a true expression.
- You would just have to follow the order of operations.
- But throwing those parentheses there makes it a little bit
- easier to read.
- Let's try 4d.
- 12 minus 8 minus 4 times 5 is equal to minus 8.
- So first let's just see what happens if we did traditional
- order of operations.
- If we did traditional order of operations we would do this 4
- times 5 first, which would give us 20 over there.
- And then we would have 12 minus 8 is 4.
- And then we would do 4 minus 20-- No, that's not right.
- That would give us negative 16.
- So that's not going to be right.
- So we can't just do traditional order of
- operations.
- Sorry, this is a minus 8 right there.
- So let's see how we can experiment with this.
- Let's try out a couple of situations.
- What if we did 12 minus 8 minus 4 and then multiplied
- that times 5.
- Let's see what this give us.
- I'm just experimenting with parentheses.
- So if you do 8 minus 4, that right there would be
- 8 minus 4 is 4.
- And then 4 times 5 would be 20, and then 12 minus 20--
- yeah, that works.
- So let me confirm that.
- So I'm saying I'm going to put parentheses right there and
- right there and let's work it out.
- You would get 8 minus 4 is 4.
- So this whole thing was simplified to 12
- minus 4 times 5.
- And you just do order of operations, you do
- multiplication first. So that is just 20.
- And if I wanted to make it very clear I could actually
- write it like this.
- I could actually put another round of
- parentheses right like that.
- But order of operations would tell us to do it anyway.
- So it becomes 12 minus 20, which is, indeed, minus 8 or
- negative 8.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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