Main content

## Algebra (all content)

### Course: Algebra (all content) > Unit 2

Lesson 12: Old school equations with Sal# Linear equations 4

Solving linear equations with variable expressions in the denominators of fractions. Created by Sal Khan.

## Want to join the conversation?

- My problem is with adding or subtracting values once you go through the motions of the first step when you get rid of the (x) in the denominator. EX: -x+2/4x+2=3 my problem is around step two I have issues with the format of setting up the adding or subtracting. Is their a rule for it. Because in this problem you would need to subtract 12x from both sides but their is a -x on the other side already! Is it suppose to be set up like 12x-x or -x-12x?(16 votes)
- -x+2/4x+2=3

-x+2=12x+6

-x-12x=6-2

-13x=4

x=-4/13

isn't that simple...

you have to just follow some rules....(6 votes)

- what if say the equation is 1/2(8x+4)=3/4(4x-6)

How would you solve that. I got a question like this on my grade 9 math quiz and I totally blacked out.(9 votes)- you would distribute the 1/2 to 8x + 4

4x+2=3/4 (4x-6)

you would then distribute the 3/4 to 4x-6

4x+2=3x-4.5

subtract 3x from both sides

1x+2= - 4.5

then subtract two from both sides

x= - 6.5(4 votes)

- Parent struggling with 8th grade homework, how do you do this problem 12(2k +11) = 12(2k+1)(4 votes)
- Laura,

You probably are working it correctly, but this equation has no solutions.

12(2k +11) = 12(2k+1) First divide both sides by 12

12/12 * (2k +11) = 12/12 * (2k+1) The 12/12 becomes 1 and disappear.

(2k +11) = (2k+1) Now subtract 2k from both sides

2k-2k+11 = 2k-2k+1 The 2k-2k becomes zero

0+11 = 0+1

11=1 This is false. So the equation has no solution. There is no value you can put in for k to solve this equation.

I hope that helps.(9 votes)

- On the exercises for this video, it is hard to know whether or not to write the answer in a fraction or a decimal. Can something be fixed about that?(6 votes)
- yeah man . jus do it in fraction form. youl get it correct(1 vote)

- Ok, so I was doing this excersise on kahn acedemy, and it was this..

-4x-3/-3x+1 = 5 I got 8/-19, but the answer was 8/11, and i really don't understand why this is so.(4 votes)- Start by multiplying both sides of the equation by -3x+1. This gives -4x-3 = -15x+5. Add 15x to both sides, giving 11x-3 = 5. Add 3 to both sides, giving 11x=8. Divide both sides by 11, giving x=8/11.

It looks like you subtracted 15 from both sides instead of adding. This gave you -19x instead of 11x.(6 votes)

- Hi Sal, I have a question on one fractional linear equation I just can't seem to solve.

(5x/3) +9= (x/6) +18

It says the answer is 6 but I just can't seem to understand where I'm going wrong with this. Can you please explain? Thanks!(3 votes)- So, it goes like this.

(5x/3) +9= (x/6) +18 \subtract 9 from both sides

(5x/3)=(x/6)+9 \subtract (x/6) from both sides

(5x/3)-(x/6)=9 \do some reorganization

(5/3)x-(1/6)x=9 \fractions to the common base

(10/6)x-(1/6)x=9

(9/6)x=9 \divide both sides by 9

(1/6)x=1 \multiply both sides by 6

x=6 \done(3 votes)

- can anyone please help me to solve this linear Equation

7(8-x/4)=77

the answer is -12 but i have tried Everything i can't get it?(2 votes)- 7(8-x/4)=77 Step1=Simplify,divide both sides of the equation by 7 :(8-x/4)=11

Step2=Open the brackets and -8 from both sides.: -x/4=3

Step3=Multiply both sides by 4 to get rid of denominator.: -x=12

Step4=Multiply both sides by -1 to get x=12 Hope this helps. :)(5 votes)

- You combine "like" terms if there are 2 variables....so how would you combine if there were fractions? eg. ( 2 3/4 x)(1 vote)
- combine all the fractions that have the same variable:)(4 votes)

- can you make a video about how to solve x+y=24 type problems(2 votes)
- There is an entire section of multiple videos on that topic. See this link: https://www.khanacademy.org/math/algebra/two-var-linear-equations(3 votes)

- At2:26, how does the "(x+1)" and "x+1" cancel out? (the problem on the left side)(2 votes)
- (x+1)

------- = 1

(x+1)

because anything divided by itself is 1. It may have been a little confusing in the video because Sal did not put parentheses around the "x+1" in the denominator. Techically it would have been better to add the parentheses when he multiplied the (x+1)/1 by the equation.

I hope that helps.(3 votes)

## Video transcript

Welcome to the presentation on
level four linear equations. So, let's start doing
some problems. So. Let's say I had the situation--
let me give me a couple of problems-- if I said 3 over x
is equal to, let's just say 5. So, what we want to do -- this
problem's a little unusual from everything we've ever seen. Because here, instead of having
x in the numerator, we actually have x in the denominator. So, I personally don't like
having x's in my denominators, so we want to get it outside of
the denominator into a numerator or at least not in
the denominator as soon as possible. So, one way to get a number out
of the denominator is, if we were to multiply both sides of
this equation by x, you see that on the left-hand side of
the equation these two x's will cancel out. And in the right side,
you'll just get 5 times x. So this equals -- the
two x's cancel out. And you get 3 is equal to 5x. Now, we could also write
that as 5x is equal to 3. And then we can think
about this two ways. We either just multiply both
sides by 1/5, or you could just do that as dividing by 5. If you multiply
both sides by 1/5. The left-hand side becomes x. And the right-hand side, 3
times 1/5, is equal to 3/5. So what did we do here? This is just like, this
actually turned into a level two problem, or actually a
level one problem, very quickly. All we had to do is multiply
both sides of this equation by x. And we got the x's out
of the denominator. Let's do another problem. Let's have -- let me say,
x plus 2 over x plus 1 is equal to, let's say, 7. So, here, instead of having
just an x in the denominator, we have a whole x plus
1 in the denominator. But we're going to
do it the same way. To get that x plus 1 out of the
denominator, we multiply both sides of this equation times x
plus 1 over 1 times this side. Since we did it on the
left-hand side we also have to do it on the right-hand
side, and this is just 7/1, times x plus 1 over 1. On the left-hand side, the
x plus 1's cancel out. And you're just left
with x plus 2. It's over 1, but we can
just ignore the 1. And that equals 7
times x plus 1. And that's the same
thing as x plus 2. And, remember, it's 7 times
the whole thing, x plus 1. So we actually have to use
the distributive property. And that equals 7x plus 7. So now it's turned into a,
I think this is a level three linear equation. And now all we do is, we say
well let's get all the x's on one side of the equation. And let's get all the constant
terms, like the 2 and the 7, on the other side of the equation. So I'm going to choose to
get the x's on the left. So let's bring that
7x onto the left. And we can do that by
subtracting 7x from both sides. Minus 7x, plus,
it's a minus 7x. The right-hand side, these
two 7x's will cancel out. And on the left-hand side
we have minus 7x plus x. Well, that's minus 6x plus
2 is equal to, and on the right all we have left is 7. Now we just have to
get rid of this 2. And we can just do that by
subtracting 2 from both sides. And we're left with minus
6x packs is equal to 6. Now it's a level one problem. We just have to multiply both
sides times the reciprocal of the coefficient on
the left-hand side. And the coefficient's
negative 6. So we multiply both sides of
the equation by negative 1/6. Negative 1/6. The left-hand side, negative
1 over 6 times negative 6. Well that just equals 1. So we just get x is equal
to 5 times negative 1/6. Well, that's negative 5/6. And we're done. And if you wanted to check it,
you could just take that x equals negative 5/6 and put it
back in the original question to confirm that it worked. Let's do another one. I'm making these up on
the fly, so I apologize. Let me think. 3 times x plus 5 is equal
to 8 times x plus 2. Well, we do the
same thing here. Although now we have two
expressions we want to get out of the denominators. We want to get x plus 5
out and we want to get this x plus 2 out. So let's do the x plus 5 first. Well, just like we did before,
we multiply both sides of this equation by x plus 5. You can say x plus 5 over 1. Times x plus 5 over 1. On the left-hand side,
they get canceled out. So we're left with 3 is equal
to 8 times x plus five. All of that over x plus 2. Now, on the top, just to
simplify, we once again just multiply the 8 times
the whole expression. So it's 8x plus 40
over x plus 2. Now, we want to get
rid of this x plus 2. So we can do it the same way. We can multiply both sides
of this equation by x plus 2 over 1. x plus 2. We could just say we're
multiplying both sides by x plus 2. The 1 is little unnecessary. So the left-hand side
becomes 3x plus 6. Remember, always distribute
3 times, because you're multiplying it times
the whole expression. x plus 2. And on the right-hand side. Well, this x plus 2 and this
x plus 2 will cancel out. And we're left with 8x plus 40. And this is now a
level three problem. Well, if we subtract 8x from
both sides, minus 8x, plus-- I think I'm running out of space. Minus 8x. Well, on the right-hand
side the 8x's cancel out. On the left-hand side we have
minus 5x plus 6 is equal to, on the right-hand side
all we have left is 40. Now we can subtract 6 from
both sides of this equation. Let me just write out here. Minus 6 plus minus 6. Now I'm going to, hope I
don't lose you guys by trying to go up here. But if we subtract minus 6 from
both sides, on the left-hand side we're just left with
minus 5x equals, and on the right-hand side we have 34. Now it's a level one problem. We just multiply both
sides times negative 1/5. Negative 1/5. On the left-hand
side we have x. And on the right-hand side
we have negative 34/5. Unless I made some careless
mistakes, I think that's right. And I think if you understood
what we just did here, you're ready to tackle some level
four linear equations. Have fun.