Prove it should have no solution

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  • Thread starter Albert1
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In summary, an equation with no solution means that there is no value or set of values that can be substituted for the variables to make it true. To prove that an equation has no solution, algebraic techniques can be used. It is possible for an equation to have no solution even if it has variables. Some examples of equations with no solution are given. An equation can also have more than one solution, with the possibility of an infinite or finite set of solutions.
  • #1
Albert1
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$a,b,c,d,e,f\in N $and all of them are odd numbers

prove :

$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}+\dfrac{1}{f}=1$ has no solution
 
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  • #2
Suppose there are odd $a, b, c, d, e, f$ satisfying the equation. Then they also satisfy:
$$bcdef + acdef + abdef + abcef + abcdf + abcde = abcdef$$
The LHS is a sum of six odd terms (products of odd numbers) so it is even, but the RHS is odd, which is a contradiction.
$\blacksquare$
 

FAQ: Prove it should have no solution

What does it mean for an equation to have no solution?

An equation with no solution means that there is no value or set of values that can be substituted for the variables in the equation to make it true. In other words, the equation has no solution because the statement is always false.

How can you prove that an equation has no solution?

To prove that an equation has no solution, we need to show that no matter what values we substitute for the variables, the equation will always be false. This can be done by using algebraic techniques such as simplifying, factoring, or using properties of equality to show that the equation cannot be true.

Can an equation have no solution even if it has variables?

Yes, an equation can have no solution even if it has variables. This means that there is no possible value that can be substituted for the variables to make the equation true. It is important to remember that the variables in an equation represent unknown values, and just because an equation has variables does not mean it has a solution.

What are some examples of equations with no solution?

Some examples of equations with no solution include:
- 2x + 5 = 2x + 8 (no value for x will make this equation true)
- x^2 + 4 = -2 (there is no real number that can be squared to equal a negative number)
- |x| = -3 (the absolute value of any number will always be positive, so there is no value for x that can make this equation true)

Is it possible for an equation to have more than one solution?

Yes, it is possible for an equation to have more than one solution. This means that there is more than one value that can be substituted for the variables to make the equation true. In some cases, an equation may have an infinite number of solutions, while in others it may have a finite set of solutions.

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