Page:Elementary algebra (1896).djvu/183

CHAPTER XIX.

Indeterminate and Impossible Problems. Negative Results. Meaning of ${\tfrac {a}{0}}$, ${\tfrac {a}{\infty }}$, ${\tfrac {0}{0}}$, ${\tfrac {\infty }{\infty }}$

179. By reference to Art. 167, it will be seen that a single equation involving two unknown quantities is satisfied by an indefinitely great number of sets of values of the unknowns involved, and that it is essential to have as many equations expressing different, or independent conditions, as there are unknown quantities to be determined. If the conditions of a problem furnish a less number of independent equations than quantities to be determined, the problem is said to be indeterminate. If, however, the conditions give us a greater number of independent equations than there are unknown quantities involved, the problem is impossible.

Suppose the problem furnishes

$3x+y=$	$10$,
$2x+y=$	$5$,
$x+y=$	$3$.

From (1) and (2) we obtain $x=5$ and $y=-5$. From (2) and (3) we obtain $x=2$ and $y=1$. These values cannot all be true at the same time, hence the problem is impossible.

NEGATIVE RESULTS.

180. A is 40 years old. and B's age three-fifths of A's. When will A be five times as old as B?

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