Question

Let $S$ be the set of all integer solutions, $(x,y,z)$, of the system of equations $x - 2y + 5z = 0 \\ - 2x + 4y + z = 0 \\ - 7x + 14y + 9z = 0$ such that $15 \leqslant {x^2} + {y^2} + {z^2} \leqslant 150$. Then, the number of elements in the set $S$ is equal to:

Answer 1

Hint: To find the number of elements in the set. We will be finding out values of the variables by first multiplying the first equation with $2$, then adding the resultant with the second equation then substitute the relation in $15 \leqslant {x^2} + {y^2} + {z^2} \leqslant 150$ to find the number of elements in the set $S$.

Complete step by step solution:
We are given system of equations
$ x - 2y + 5z = 0 \\ - 2x + 4y + z = 0 \\ - 7x + 14y + 9z = 0 $
On multiplying first equation with $2$and then adding it with the second equation, we get
$2(x - 2y + 5z) + ( - 2x + 4y + z) = 0 \\ 2x - 4y + 10z - 2x + 4y + z = 0 \\ 11z = 0 \\ z = 0 $
On substituting the value of $z$in third equation, we get
$ - 7x + 14y + 9 \times 0 = 0 \\ - 7x = - 14y \\ x = 2y $
On substituting the above obtained values and relation in condition $15 \leqslant {x^2} + {y^2} + {z^2} \leqslant 150$, we get
$ 15 \leqslant {(2y)^2} + {y^2} + 0 \leqslant 150 \\ 15 \leqslant 4{y^2} + {y^2} \leqslant 150 \\ 15 \leqslant 5{y^2} \leqslant 150 \\ 3 \leqslant {y^2} \leqslant 30 $
From this we get the value of $y$as
$\sqrt 3 \leqslant y \leqslant \sqrt {30} $which means that
$ y = [ - \sqrt {30} , - \sqrt 3 ] \cup [\sqrt 3 ,\sqrt {30} ] \\ \approx [ - 5.477, - 1.732] \cup [1.732,5.477]$
This means that the set $S$ contains elements $ \pm 2, \pm 3, \pm 4, \pm 5$.

Therefore number of elements in set is 8.

Note: There is some uncertainty here about whether or not to include obtained values of variables $(x,y,z)$ in the set. In this case, the condition $15 \leqslant {x^2} + {y^2} + {z^2} \leqslant 150$, is given, $\leqslant$ which means we must include the values obtained through this condition in our set.