HallsofIvy said:Looks straightforward. If there exist a number, x, such that
$x^2- abx+ (a+ b)= 0$ and
$x^2- (a+ b)x+ ab= 0$
then, subtracting, $(a- ab+ b)x+ (a- ab+ b)= 0$ which has the single solution $x= -1$. Putting $x= -1$ into both equations, $1+ ab+ a+ b= 0$ and $1+ a+ b+ ab= 0$. Since a and b are both positive numbers, that is impossible.
Proving no common solution means showing that there is no solution that satisfies all the given equations or statements. This is often used in mathematics to show that a set of equations or statements is inconsistent.
Proving no common solution is important because it allows us to identify inconsistencies in our equations or statements. It can also help us avoid making incorrect assumptions or conclusions based on faulty data or reasoning.
Proving no common solution often involves using mathematical techniques such as substitution, elimination, or contradiction. It may also require logical reasoning and critical thinking skills.
Yes, there can be exceptions to "proving no common solution" in certain scenarios. For example, some equations or statements may have multiple solutions that are all valid, but there is no one solution that satisfies them all simultaneously.
Yes, "proving no common solution" has real-world applications in fields such as mathematics, physics, and engineering. It can help identify inconsistencies in data or models, which can lead to more accurate predictions and solutions.