Equation Without a Solution

[nycmathguy]

Why does the following equation have no solution?

 

[MathLover1]

(3x)/(x-1)+2=3/(x-1)

(3x)/(x-1)+2(x-1)/(x-1)=3/(x-1)

(3x+2(x-1))/(x-1)=3/(x-1)

(3x+2x-2)/(x-1)=3/(x-1).......since denominator cannot be zero, x=1 is excluded from domain

then
3x+2x-2=3
5x-2=3
5x=3+2
5x=5
x=1...........since x=1, cannot be solution because it is excluded from domain

so, No Solution

 

[nycmathguy]

(3x)/(x-1)+2=3/(x-1)

(3x)/(x-1)+2(x-1)/(x-1)=3/(x-1)

(3x+2(x-1))/(x-1)=3/(x-1)

(3x+2x-2)/(x-1)=3/(x-1).......since denominator cannot be zero, x=1 is excluded from domain

then
3x+2x-2=3
5x-2=3
5x=3+2
5x=5
x=1...........since x=1, cannot be solution because it is excluded from domain

so, No Solution

An equation without a solution can be defined as empty set. Yes?

If an equation has no solution it means that it has no roots. Yes?

 

[Country Boy]

An equation without a solution can be defined as empty set. Yes?

I am not crazy about the wording here. An equation is NOT a set!

You mean that the set of all solutions to this equation (its "solution set") is empty.

If an equation has no solution it means that it has no roots. Yes?

 

[MathLover1]

An equation without a solution can be defined as empty set. Yes?

If an equation has no solution it means that it has no roots. Yes?

If an equation has no solutions, we write ∅ for the solution set. ∅ means the null set (or empty set). Sometimes, you may be given a replacement set, and asked to test whether the equation is true for all values in the replacement set.
it means that it has no roots

 

[nycmathguy]

I am not crazy about the wording here. An equation is NOT a set!

You mean that the set of all solutions to this equation (its "solution set") is empty.

Some textbooks, if memory serves me right, use the words EMPTY SET referring to an equation without a solution(s). I also don't agree or like the wording. Perhaps the way you phrased it is the correct statement found in certain textbooks.

 

[nycmathguy]

If an equation has no solutions, we write ∅ for the solution set. ∅ means the null set (or empty set). Sometimes, you may be given a replacement set, and asked to test whether the equation is true for all values in the replacement set.
it means that it has no roots

By no roots you mean no solutions, no values. Yes?