Question

The number of integral solutions to the equation is
A. 0
B. 1
C. 2
D. none of these

Answer 1

Hint: Few rules to find number of integral solutions of equations.
1. First, reduce the equation in lowest reducible form.
2. After reducing, if coefficients of x and y still have a common factor, the equation will have no solutions.
3. If x and y are co-prime in the lowest reducible form, find any one integral solution. The rest of the
    solutions can be derived from that integral solution.
4. For each successive integral solutions of the equation, the value x and y will change by a coefficient of
    the other variable. If the equation is of the type Ax – By=C (after getting the lowest reducible form), an
    increase in x will cause increase in y. If the equation is of the type Ax + By=C, an increase in x will cause
    a decrease in y.

Complete step by step solution: For integral solution,
$x + y - [x][y] = 1$
Simplify the given equations to find the values of “x” and “y”
$x + y - xy = 0$
Subtracting minus one from both the sides
$ - 1 + x + y - xy = ( - 1)$
Taking factorization –
\[\begin{array}{l}
 - 1 + x + y - xy = - 1\\
\underline {1 - x} - \underline {y + xy} = 1\;{\rm{ (Taking - 1 common from both the sides)}}\\
1(1 - x) - y(1 - x) = 1
\end{array}\]
This is only possible if
$\begin{array}{l}
1 - x = 1\\
1 - x = 1\\
\therefore x = 0\\
{\rm{ }}
\end{array}$
Similarly if,
 $\begin{array}{l}
1 - x = - 1\\
x = 2{\rm{ ,}}\\
{\rm{1 - y = - 1}}\\
{\rm{y = 2}}
\end{array}$
∴ Solutions are $(0,0)\;{\rm{or (2,2)}}$
Hence, from the given multiple choices, option C is the correct answer.

Note: An Integral solution is a solution such that all the unknown variables take only integer values. Given three integers a, b, c representing a linear equation of the form: ax + by = c. Determine if the equation has a solution such that x and y are both integral values.