Solve Rational Equation (y-1)/(1-y)=1

Published on October 10, 2026

Algebra

Determine that the rational equation (y-1)/(1-y)=1 has no solution because the denominator is the negative of the numerator, making the left side equal -1.

Question

Answer

Factor the denominator:

Substitute:

So the equation becomes:

This is a contradiction, so the equation has no solution.

Theorems

• Additive Inverse Property

For any real number

, the sum of

and its additive inverse is zero. In algebra, this allows factoring out

from a binomial:

• Zero Denominator Rule

A rational expression is undefined if its denominator equals zero. Therefore, the domain restriction for the equation is

, which implies

• Inconsistent Equation

An equation that simplifies to a false statement (contradiction) regardless of the variable’s value has no solution. In this case, simplifying the equation yields

, which is false, so no solution exists.

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