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## Algebra (all content)

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

Lesson 9: Analyzing the number of solutions to linear equations- Number of solutions to equations
- Worked example: number of solutions to equations
- Number of solutions to equations
- Creating an equation with no solutions
- Creating an equation with infinitely many solutions
- Number of solutions to equations challenge

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# Worked example: number of solutions to equations

Sal attempts to solve 8(3x + 10) = 28x - 14 - 4x only to find that the equation has no solution. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- did i do something wrong i only got the number left

Solve for X

8(3x + 10) = 28x โ 14 โ 4x

Multiplied the numbers in the brackets

24x + 80 = 28x โ 14 โ 4x

Compressed the 2 โxโ numbers left side

24x + 80 = 24x โ 14

got rid of the 14 on the right by add 14 to both sides and Took away the 24 by x on both sides

94(10 votes)- 24๐ฅ โ 24๐ฅ + 94 = 24๐ฅ โ 24๐ฅ

0 + 94 = 0

94 = 0

This is of course a false statement and means that the equation doesn't have a solution.

No matter what value of ๐ฅ we put into the equation we will never get equality.(18 votes)

- If you are solving a problem and you come to one of these special cases, can you just write null set? The symbol is like a 0 with a line crossing diagonally through it. On a test just drawing the symbol might be easier and less time consuming, hence my question.(3 votes)
- I'm not exactly sure if you can use that, some teachers may agree other may not. To be on the safer side I recommend that you say 'No Solution' as an answer. Since we couldn't find what 'x' was exactly there wasn't a solution to the answer.

I hope this helped.(8 votes)

- why are functions so hard for me even though that we have studied them for weeks(5 votes)
- Not sure. Have you considered asking a teacher to try to explain?(6 votes)

- is this actually a equation at all? I'm not sure(5 votes)
- then the equation has infinite solutions(3 votes)

- Guys, I think Sal attempts to solve 8(3x + 10) = 28x - 14 - 4x only to find that the equation has no solution.(4 votes)
- 3-3=0

2-2=0

Concludion: 2=3(1 vote)- You have a false conclusion.

The inverse property of addition tells us that if you subtract a number from itself, it will always = 0.

It does not tell you that if you apply the property multiple times that the 2 numbers you used will equal each other.

2 will never equal 3.(6 votes)

- I think Sal was talking way too fast, I didn't really get the term.(2 votes)
- If there is no solution then does that mean that there could be other answers ?(4 votes)

- HI!

I just want to ask, At1:28what do Sal meant by these guys(2 votes)- 24x and -24x on the left hand side of the equation(3 votes)

- Wait, stupid question, why can't we subtract the 80 first, and then divide 24x by 24?(2 votes)
- You could subtract 80 first. But, dividing by 24 doesn't help you solve for X. You need to get all the X's on the same side. So, at some point, you need to subtract the X's. If you divide, you are just making extra work by introducing a step that is not needed.

Hope this helps.(3 votes)

- What if the equation simplifies to something like 5x equals 3x? Would zero count as a solution to this equation, making the number of solutions that this equation has 1?(3 votes)

## Video transcript

Solve for x. We have 8 times the
quantity 3x plus 10 is equal to 28x
minus 14 minus 4x. So like every equation
we've done so far, we just want to isolate
all of the x's on one side of this equation. But before we do
that, we can actually simplify each of these sides. On the left-hand
side, we can multiply the quantity 3x plus 10 times 8. So we're essentially
just distributing the 8, the distributive
property right here. So this is the same thing
as 8 times 3x, which is 24x, plus 8 times 10, which
is 80, is equal to-- and over here, we have
28x minus 14 minus 4x. So we can combine the
28x and the minus 4x. If we have 28x minus
4x, that is 24x And then you have the
minus 14 right over here. Now, the next thing we
could-- and it's already looking a little bit suspicious,
but just to confirm that it's as suspicious as
it looks, let's try to subtract 24x from both
sides of this equation. And if we do that, we see that
we actually remove the x's from both sides of the equation
because we have a 24x there, and we have a 24x there. You might say,
hey, let's put all the x's on the left-hand side. So let's get rid of this 24x. So you subtract 24x
right over there, but you have to do it to
the left-hand side as well. On the left-hand side,
these guys cancel out, and you're left with just
80-- these guys cancel out as well-- is equal
to a negative 14. Now, this looks very bizarre. It's making a statement that
80 is equal to negative 14, which we know is not true. This does not happen. 80 is never equal
to negative 14. They're just inherently inequal. So this equation right here
actually has no solution. This has no solution. There is a no x-value that will
make 80 equal to negative 14.