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## Algebra (all content)

# Linear equations 1

To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation. Created by Sal Khan.

## Video transcript

Welcome to level one
linear equations. So let's start
doing some problems. So let's say I had the equation
5-- a big fat 5, 5x equals 20. So at first this might look
a little unfamiliar for you, but if I were to
rephrase this, I think you'll realize this
is a pretty easy problem. This is the same thing as
saying 5 times question mark equals 20. And the reason we do the
notation a little bit-- we write the 5 next
to the x, because when you write a number right
next to a variable, you assume that you're
multiplying them. So this is just
saying 5 times x, so instead of a question
mark, we're writing an x. So 5 times x is equal to 20. Now, most of you all could
do that in your head. You could say, well, what
number times 5 is equal to 20? Well, it equals 4. But I'll show you a way to do
it systematically just in case that 5 was a more
complicated number. So let me make my pen
a little thinner, OK. So rewriting it, if
I had 5x equals 20, we could do two things
and they're essentially the same thing. We could say we just divide
both sides of this equation by 5, in which case, the left
hand side, those two 5's will cancel out, we'll get x. And the right hand side,
20 divided by 5 is 4, and we would have solved it. Another way to do it, and this
is actually the exact same way, we're just phrasing
it a little different. If you said 5x equals 20,
instead of dividing by 5, we could multiply by 1/5. And if you look at that, you can
realize that multiplying by 1/5 is the same thing
as dividing by 5, if you know the difference
between dividing and multiplying fractions. And then that gets the
same thing, 1/5 times 5 is 1, so you're just
left with an x equals 4. I tend to focus a
little bit more on this because when we start having
fractions instead of a 5, it's easier just to think about
multiplying by the reciprocal. Actually, let's do one
of those right now. So let's say I had negative
3/4 times x equals 10/13. Now, this is a harder problem. I can't do this one in my head. We're saying negative
3/4 times some number x is equal to 10/13. If someone came up to you on
the street and asked you that, I think you'd be like me,
and you'd be pretty stumped. But let's work it
out algebraically. Well, we do the same thing. We multiply both sides
by the coefficient on x. So the coefficient, all that
is, all that fancy word means, is the number that's
being multiplied by x. So what's the
reciprocal of minus 3/4. Well, it's minus 4/3 times,
and dot is another way to use times, and
you're probably wondering why in algebra, there
are all these other conventions for doing times
as opposed to just the traditional
multiplication sign. And the main reason is, I think,
just a regular multiplication sign gets confused
with the variable x, so they thought of either using
a dot if you're multiplying two constants, or just writing
it next to a variable to imply you're
multiplying a variable. So if we multiply the left
hand side by negative 4/3, we also have to
do the same thing to the right hand
side, minus 4/3. The left hand side, the
minus 4/3 and the 3/4, they cancel out. You could work it out on
your own to see that they do. They equal 1, so we're just
left with x is equal to 10 times minus 4 is minus 40, 13 times
3, well, that's equal to 39. So we get x is equal
to minus 40/39. And I like to leave
my fractions improper because it's easier
to deal with them. But you could also view
that-- that's minus-- if you wanted to write it
as a mixed number, that's minus 1 and 1/39. I tend to keep it like this. Let's check to make
sure that's right. The cool thing about algebra is
you can always get your answer and put it back into
the original equation to make sure you are right. So the original equation
was minus 3/4 times x, and here we'll substitute
the x back into the equation. Wherever we saw x, we'll
now put our answer. So it's minus 40/39, and
our original equation said that equals 10/13. Well, and once
again, when I just write the 3/4 right next to
the parentheses like that, that's just another
way of writing times. So minus 3 times minus
40, it is minus 100-- Actually, we could do
something a little bit simpler. This 4 becomes a 1
and this becomes a 10. If you remember when you're
multiplying fractions, you can simplify it like that. So it actually becomes
minus-- actually, plus 30, because we have a minus times
a minus and 3 times 10, over, the 4 is now 1, so all
we have left is 39. And 30/39, if we divide the
top and the bottom by 3, we get 10 over 13, which
is the same thing as what the equation said
we would get, so we know that we've got
the right answer. Let's do one more problem. Minus 5/6x is equal to 7/8. And if you want to
try this problem yourself, now's a
good time to pause, and I'm going to start
doing the problem right now. So same thing. What's the reciprocal
of minus 5/6? Well, it's minus 6/5. We multiply that. If you do that on
the left hand side, we have to do it on the
right hand side as well. Minus 6/5. The left hand side, the
minus 6/5 and the minus 5/6 cancel out. We're just left with x. And the right hand
side, we have, well, we can divide both
the 6 and the 8 by 2, so this 6 becomes negative 3. This becomes 4. 7 times negative
3 is minus 21/20. And assuming I haven't
made any careless mistakes, that should be right. Actually, let's just
check that real quick. Minus 5/6 times minus 21/20. Well, that equals
5, make that into 1. Turn this into a 4. Make this into a 2. Make this into a 7. Negative times
negative is positive. So you have 7. 2 times 4 is 8. And that's what we
said we would get. So we got it right. I think you're
ready at this point to try some level one equations. Have fun.