To solve the system of equations using the elimination method, we start by writing down the given equations:
[tex]\[
\frac{1}{x} + \frac{1}{y} = 5 \quad \text{(1)}
\][/tex]
[tex]\[
\frac{1}{y} + \frac{1}{x} = 1 \quad \text{(2)}
\][/tex]
We observe that equations (1) and (2) are identical. Thus, these equations are in essence saying the same thing and represent a single line in the xy-plane, not two distinct lines. Let us be clear that the system has no unique solution because the two equations are dependent and repeat the same relationship, i.e.,
[tex]\[
\frac{1}{x} + \frac{1}{y} = 5 \quad \text{and} \quad \frac{1}{x} + \frac{1}{y} = 1
\][/tex]
Since the equations effectively represent the same relationship, and since [tex]\(5 \neq 1\)[/tex], this system is contradictory and inconsistent. There are no values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that can satisfy both equations simultaneously. Therefore, the system has no solution.
