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### Unit 4: Lesson 6

Lesson 7: All, some, or no solutions

# Creating an equation with no solutions

Sal shows how to complete the equation -11x + 4 = __x + __ so that it has no solutions. Created by Sal Khan.

## Want to join the conversation?

• How do I find the value of a constant, such as (k) where there are no solutions? How would I solve it if the equation 4(80 + n) = (3k)n ?
• I think you are saying that you need to find a value of "k" so that the equation will have no solution.
For this to happen...
1) the coefficient of "n" must match on both sides of the equation
2) the constant on each side must be different.

Start by simplifying your equation -- distribute the 4: 320 + 4n = 3kn
The constants on each side are different: 320 on left, and 0 on right. So, one condition is met.
We now know that the coefficient of "n" must = 4. You can find "k" by setting 3k = 4 and solving for "k".

Hope this helps.
• Is there any simple trick to find the equation which has no solution without even solving it
• This trick is based on simplifying and as soon as you see the same coefficients of the variable on both sides and any different numbers on the two sides, you know that there are no solutions.
Example: 2(2x+7)= 5x +12 -x
Distribute on left to get 4x +14
Combine like terms on right to get 4x + 12
Since the coefficients of x are both 4, but the constants are different, you know there are no solutions because if you took it to the end, you would get 2=0 which can never be true.
(1 vote)
• how many solution does 4(2x – 3) – 5 = 4x have
• The overall coefficient on x on the left-hand side is 4*2=8.
The overall coefficient on x on the right-hand side is 4.
These coefficients are unequal, so this linear equation has exactly one solution.
• why would we want to make a linear equation that has no solution, isn't the point of an equation to solve for the variable?
• Wow, this comment is old, where are you now? anyway, my guess is that by creating a linear equation with no solution at all we could instantly distinguish equations with no solution with just 1 step or less and thus saving time.
• Is there any real world application for making an equation with no solution?
(1 vote)
• No, there can't be, because it wouldn't exist. If there is no solution, there can't be an existance.
• Wait. Equations with no solution cannot apply to something in real life because of the laws of thermodynamics, so if these equations have no real life use why are we learning about them at all. Or do they have a real life use.
• That is actually almost true, but the reason we learn them is to show that there are equations with no answer, but yes there is no real life application since you will never in real life with real problems ever really experience something with no solution.
• how do we find the equations with one solution
• The linear equations with exactly one solution are precisely those that have different overall coefficients on x on the two sides.
• For example in the inequality 75x+57= -75x+57 you end up with 150x=0. When it's equal to zero how many solutions does it have??
(1 vote)
• 75𝑥 + 57 = −75𝑥 + 57 is an equation that simplifies to 150𝑥 = 0

Dividing both sides by 150, we get 𝑥 = 0, which is the only solution.