If I use the intercept form or simply find the slope by dividing the $y$-intercept by the $x$-intercept then the slope will be equal to $1$. But logically there must be two lines satisfying the given condition as seen in the graph below, the other having slope equal to $-1$. How do I mathematically show the existence of the other line and why the method I used only a gave one answer.
The purple line is not a solution: the $y$ and $x$ intercepts have opposite signs.
Generally intercept means the length between intersection of axis and line and the origin, with sign. So there is only one line which satisfies this.