Equations for beginners
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Variables Expressions and Equations
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Simple Equations
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Solving One-Step Equations
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Solving One-Step Equations 2
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One Step Equations
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One step equations
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One-step equations with multiplication
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Two-Step Equations
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Solving Ax+B = C
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Example: Dimensions of a garden
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2-step equations
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Example: Two-step equation with x/4 term
Variables Expressions and Equations Introduction and examples of variables, expressions and equations
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- When we're dealing with basic arithmetic
- we see the concrete numbers there.
- We'll see 23 + 5.
- We know what these numbers are right over here
- and we can calculate them.
- It's going to be 28.
- We can say 2 x 7.
- We could say 3 / 4.
- And all of these cases we know exactly
- what numbers we're dealing with.
- As we start entering into the Algebratic world
- and you probably have seen this a little bit already.
- We start dealing with the ideas of variables.
- And variables, there's a bunch of ways
- you can think about them
- but they're really just values and expressions
- where they can change.
- The values in those expressions can change.
- So for example, if I write, if I write,
- x + 5
- this is an expression right over here.
- This can take on some value
- depending on what the value of x is.
- If x is equal to, so if x is equal to 1,
- then x + 5, our expression right over here.
- Is going to be equal to 1.
- Because now x is 1.
- It'll be 1 + 5.
- So x + 5 will be equal to 6.
- If x is equal to, I don't know, -7
- then x plus 5, is going to be equal to,
- well now x is -7.
- It's going to be -7 + 5 which is negative 2.
- So notice.
- x here is a variable, x here is the variable,
- and its value can change depending on the context.
- And this is in the context of an expression.
- You'll also see that in the context of an equation.
- It's actually important to realize this distinction
- between the expression and an equation.
- An expression is really just a statement of values
- a statement of some type of quantity.
- So this is an expression.
- An expression would be something like.
- Well, what we saw over here.
- x + 5
- the value of this expression will change
- depending on what the value of this variable is.
- And you could just evaluate for different values of x
- Another expression could be something like
- I don't know y + z.
- Now everything is a variable.
- If y is 1 and z is 2,
- it's going to be 1 + 2.
- If y is 0 and z is -1
- it's going to be 0 + (-1).
- These can all be evaluated
- and they'll essentially give you a value
- depending on the values of each of these variables
- that make up the expression.
- An equation, you're essentially setting expressions
- to be equal to each other.
- that's why they're called 'equations'
- you're equating two things.
- an equation you'll see one expression
- being equal to another expression.
- So for example, you could say something like...
- x + 3 = 1
- and in this situation where you have an equation
- where you have one equation with only one unknown.
- You could actually figure out
- what x needs to be in this scenario.
- and you could might even do it in your heads.
- What plus 3 is equal to 1?
- well you can do that in your head.
- if I have -2 + 3 is equal to 1.
- so in this context an equation is starting to constrain
- what value that this variable can take on.
- but it doesn't have necessarily constrain as much.
- You could have something like,
- x + y + z = 5
- now you have this expression that is
- equal to this other expression.
- 5 is really just an expression right over here.
- and there are some constraints.
- If someone tells you what y and z is
- and you're going to get an x.
- If someone tells you what x and y is
- and that constrains what z is.
- But it depends on what the different things are.
- So for example
- if we said y = 3, and z = 2
- then what would be x in that situation?
- so if y = 3, and z =2
- then you're going to have
- the left hand expression is going to be
- x + 3 + 2
- is going to be x + 5
- this part right over here is going to be 5
- x + 5 = 5
- and so what + 5 = 5?
- well now we're constraining that
- x would have to be...
- x would have to be equal to 0
- But the important point here, one
- you hope you realize the difference
- between an expression and an equation.
- an Equation is essentially
- you're equating two expressions.
- the important thing to take away from here,
- is that a variable can take on different values
- depending on the context of the problem.
- And to hit the point home,
- let‘s just evaluate a bunch of expressions,
- when the variables have different values.
- So for example, if we had the expression
- if we had the expression,
- x to the... x to the y power
- if x is equal to... if x is equal to 5
- and y is equal to 2
- y is equal to 2.
- then our expression here is going to evaluate to...
- well x is now going to be 5.
- x is going to be 5.
- y is going to be 2.
- it's going to be 5 to the second power.
- or it's going to evaluate to
- 25.
- If we change the values,
- if we said, x... if we said,
- let me do that in the same colour.
- If we said x is equal to... x is equal to -2
- and y... and y is equal to 3
- then this expression would evaluate to,
- it would evaluate to, let me do in that (colour)
- so it would evaluate to
- -2, that's what we're going to substitute for x now
- in this context.
- and y is now 3
- -2 to the third power... -2 to the third power
- which is -2 x -2 x -2
- which is -8
- -2 x -2 = +4
- x -2 again is equal to -8
- is equal to -8
- so you see depending on what the values of these are
- you know we could even do more complex things.
- We could have an expression like,
- the square root of x + y and then minus x, like that.
- If x is equal to, let's say that x is equal to 1
- and y... y is equal to 8
- then this expression would evaluate to
- well every time we see an x we wanna put a 1 there.
- so we would have a 1 there.
- and you would have a 1 over there.
- And every time you would see a y,
- you would put an 8 in its place.
- And in this context, we're setting these variables
- so you would see an 8.
- So under the radical sign you would have a 1+8.
- So you would have the principal root of 9, which is 3
- So this whole thing would simplify in this context.
- When we set these variables to be these things,
- this whole thing would simplify to be 3.
- 1 plus 8 is 9
- principal root of that is 3
- and then you would have 3 - 1
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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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