WEBVTT
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Hi, this is Anthony Varela.
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And today, we're going to be
solving single-step equations.
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So what we're going to do is
talk about rules in equations.
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Then we're going to look
at inverse operations
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and how that can help
us solve equations,
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and then we're going to
practice solving equations
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in a single step.
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So let's talk about equations.
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Well, here is an equation
x plus 13 equals 20.
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And what an equation is is
it's a mathematical statement
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that expressions or quantities
have the same value.
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So this is saying that
the quantity x plus 13 has
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the same value as 20.
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So we have this
variable x in here.
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And a variable is a
value that can change.
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But when it's in
an equation, there
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is then a certain value that
makes the statement true.
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Then x could have several
values that make it true.
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But it's not always
the case that x
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can be whatever it wants to be.
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It has to fit this equation.
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And another thing
about equations
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is that I can write this
in the other direction.
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I could say 20 equals x plus 13.
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And this is what we call
the rule of symmetry,
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that you can write the equation
in the other direction.
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So with equations,
we have quantities
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that have the same value.
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And we can read this
forwards or backwards.
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So now, let's talk about
isolating the variable, which
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is the same thing as
solving for your variables
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in your equation.
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So basically, what
you want to do
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is you want to
rewrite this equation
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so that it looks
something like this.
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We have our variable x on
one side of the equation,
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and then everything else is
going to be on the other side
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that we can evaluate.
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So what we're
asking ourselves is
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what's happening
to x, our variable
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that we'd like to solve for.
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And we see that we're adding 13.
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So how can we undo
this addition of 13?
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Well, we can subtract 13.
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And this is using
inverse operations.
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So what we're going
to do is subtract 13
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from this left side
of the equation that's
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going to cancel out or
undo 13 being added to x.
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But what we have to do on the
other side of the equation
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is subtract 13 as well.
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And this is called
the rule of equality.
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And you've probably heard
your math teacher say
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this many, many, many times.
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What you do to one side,
you must do to the other.
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They're referring to
the rule of equality.
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So we have x plus 13.
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Then we take away 13.
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So we have just x on one
side of our equation.
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And on the other side of the
equation, we have 20 minus 13,
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which is 7.
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So now, you've
solved this equation.
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We can plug 7 in for x and
say, OK, 7 plus 13 equals 20.
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I've solved my equation.
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So let's talk about
inverse operations
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because we're going to be using
these to solve our equations.
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We've already talked about
addition and subtraction
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being inverse operations.
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So here's an
example that we just
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walked through where if we
add 13, we could subtract 13.
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Now, subtraction is the
inverse operation of addition.
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So if we have x minus 13,
we can add 13 to undo that.
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Multiplication and division
are inverse operations.
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So here's an example where we
have 2 being multiplied by x.
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So we divide by 2 to undo that.
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And division is the inverse
operation of multiplication.
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So if we have x divided
by 2, we can multiply by 2
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to get back to x.
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Now, squaring and
taking the square root
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are inverse operations.
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So if we have a positive
number x squared,
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we can take the square root
of that to get back to x.
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And similarly, if we
have a positive number
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x that we have the
square root of that,
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we can square that
and get back to x.
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And we can generalize
squaring and taking
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the square root to talking
about the n-th power
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and the n-th root as
being inverse operations.
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So here are our
inverse operations--
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addition and subtraction,
multiplication and division,
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and then we have
powers and roots.
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So now, let's go through
a whole bunch of examples
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where we're using
these different inverse
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operations to solve
equations in a single step.
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So here, we have 7/3
equals x minus 2/3.
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So looking at the operation that
I see next to that variable x,
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I see subtracting 2/3.
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So we need to add 2/3 applying
the inverse operation.
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And what we do to one side,
we have to do to the other.
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So now, our equation
is 7/3 plus 2/3,
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which is 9/3 equals
x and 9/3 equals 3.
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So that's how we can
solve that equation.
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Our next equation
is 5 equals 2x.
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And what operation
do I see here?
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Well, I see multiplication.
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2x is 2 times x.
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So I need to apply the inverse
operation, which is division.
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So I need to divide by 2 on
both sides of my equation.
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So x equals 5/2.
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And in the next example,
we have x over 3 equals 8.
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Now, what operation
do I see here?
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I see x being divided by 3.
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So I need to apply the
inverse operation of division,
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which is multiplication.
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So I need to multiply
by 3 on the left side
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and multiply it by
3 on the right side.
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So now, I have x
equals 8 times 3 or 24.
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In this example, we
have 8 equals x squared.
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And then what operation
do I see here?
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I see x being squared.
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So I need to take
the square root,
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applying the inverse operation.
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And then I need to do this on
the other side of the equation
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as well.
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So now, what do I get then?
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Well, the square root
of x squared equals x.
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And then we have the
square root of 8.
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And now, it's important when
you're taking square roots
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that you're going to
include the plus or minus
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when we're talking
about equations
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because negative
root 8 times negative
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root 8 will also give us 8.
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And our last example
here, we have 7.8
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equals the square root of x.
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So how we're going to
solve this equation-- well,
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I see that we're taking
the square root of x.
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So we're going to square
that side of the equation.
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Now, since we squared the
right side of the equation,
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we have to square
the other side.
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So x equals 7.8
squared, which is 60.84.
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So let's review solving
a single-step equation.
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Well, we talked about
equations being quantities
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that have the same value,
that statement that quantities
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had the same value.
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We could read equations
both forwards and backwards.
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The rule of inequality says
any operation performed
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on one side of the
equation must be performed
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on the other side
in order to keep
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the quantities equal in value.
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That's important.
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And to solve our equations,
we applied inverse operations.
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Addition and subtraction
are inverse operations.
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Multiplication and division
are inverse operations.
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And powers and roots
are inverse operations.
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Well, thanks for watching this
video on solving single-step
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equations.
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Hope to see you next time.