### Video Transcript

Let’s talk about solving equations
with rational numbers. Before we do that, we need to talk
about a few key concepts. The first one is the reciprocal;
it’s also called the multiplicative inverse. The reciprocal is whatever you
multiply a number by and the product becomes one. It’s represented here by 𝑎 times
one over 𝑎. When you multiply those together,
𝑎 over 𝑎 equals one. five times one-fifth equals one. one-fifth is the reciprocal
or multiplicative inverse of five.

Next step, the addition property of
equality. This property states that if 𝑎
equals 𝑏, then 𝑎 plus 𝑐 equals 𝑏 plus 𝑐. Here’s an example of using the
addition property of equality to solve an equation. I’ve added five to both sides here
to solve for 𝑥. Sometimes we might say if you do
something to one side of the equation then you have to do the same thing to the
other. When we say this, we are using the
addition property of equality.

Our third and final key concept,
the multiplication property of equality which is similar to the addition property of
equality. It states if 𝑎 equals 𝑏 then 𝑎
times 𝑐 equals 𝑏 times 𝑐. Here’s our example: 𝑎 divided by
five equals seven. To solve for 𝑎, I’ll multiply by
five on both sides of the equation, and I’m using the multiplication property of
equality to do this.

Now on to solving equations.

Example one: one-half 𝑥 equals
negative five-sixths. The goal here is to solve for
𝑥. And we need to do that by first
using the reciprocal to get 𝑥 by itself. Here’s what that looks like: I
multiply both sides by two over one. Once I do that, I get 𝑥 equals
negative ten-sixths. But this is not the simplified form
of this fraction, and we always wanna keep the fractions in simplest form. The simplified form is negative
five-thirds. To find this, I divided negative
ten-sixths by two on the top and two on the bottom.

Next, 𝑥 minus one-seventeenth
equals five-seventeenths. We’re gonna solve this one with the
addition property of equality. By adding one-seventeenths to both
sides of the equation, our final answer becomes 𝑥 equals six-seventeenths.

Example three: when three-fourths
is divided by 𝑎 over 𝑏, the result is five-eighths. Solve for 𝑎 over 𝑏. The first step is to turn this word
problem into an equation. We’ve done that here by writing
three-fourths divided by 𝑎 over 𝑏 equals five-eighths. Don’t forget! When we divide by a fraction, that
means we’re multiplying by the reciprocal. So that’s what we’re going to do;
we’re going to change the division to multiplication and change the 𝑎 over 𝑏 to 𝑏
over 𝑎. Just gonna slide the problem up so
we can keep going. We’re trying to get 𝑏 over 𝑎 by
itself, so I multiplied by four-thirds on both sides. After multiplying by four-thirds on
both sides, we’re left with 𝑏 over 𝑎 equals twenty over twenty-four. Again, we always want the most
simple form that we can find, so I know that twenty over twenty-four can be
reduced. A simplified form of that fraction
is five-sixths; we divided the top and the bottom by four. So we found 𝑏 over 𝑎, but our
question wasn’t asking us what 𝑏 over 𝑎 was. Our question was asking us to solve
for 𝑎 over 𝑏, so we flip our fraction for the final answer, and 𝑎 over 𝑏 equals
six-fifths.

The next question says three and
seven-thirteenths divided by some number equals one. Three and seven-thirteenths divided
by some number equals one; I just want you to think about this problem for a
minute. What do you think should go
there? How do you think we should solve
this problem? If you’re still not sure, here’s a
hint: three divided by what equals one, or five divided by what equals one. Itself! three divided by three
equals one and five divided by five equals one. Anything divided by itself equals
one. We only had to remember that fact
to answer this question. The answer here is three and
seven-thirteenths because three and seven-thirteenths divided by itself equals
one.

Example five: fourteen
twenty-sevenths divided by some number equals one and five-sixths, so we get
fourteen twenty-sevenths divided by 𝑎 equals eleven-sixths. Now I’ve have made two changes:
I’ve changed divided by 𝑎 into multiplied by one over 𝑎. So we went from dividing by
something to multiplying by its reciprocal. In order to isolate our variable, I
multiplied by the reciprocal twenty-seven fourteenths on both sides of the
equation. To simplify, I divided twenty-seven
by three and six by three, and I got nine and two. This will help finding the simplest
form of this fraction. Now we’re left with one over 𝑎
equals eleven times nine over two times fourteen. When we multiply that out, we get
one over 𝑎 equals ninety-nine over twenty-eight. But if you look closely, you’ll see
that we’re not looking for one over 𝑎; we’re actually looking for 𝑎. So our final answer here has to be
twenty-eight over ninety-nine.

As you solve these problems,
remember the key concepts. They are your tools for solving
problems like these. Your reciprocal, being 𝑎 times one
over 𝑎 equals one. The addition property of equality,
if you add something to one side, you have to add it to the other. And the same goes for
multiplication, if you multiply by something on one side, you have to do the same
thing on the other. And now you’re ready, so it’s your
turn to try some.