Question

Let $\left( x,y,z \right)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x-y-z=0$, $-3x+z=0$, $-3x+2y+z=0$. Then the number of such points which lie inside a sphere of radius $10$ centered at the origin is

Answer 1

For this problem we need to calculate the number of points which satisfies the given coordinate system and the given condition. For this we will first consider the given coordinate system and solve those equations. By observing the equation, we can simply add first and second equations to get the value of $y$. Now we will substitute this value in the third equation to get the value of $z$ in terms of $x$. By assuming the value of $x$ to a constant say $k$ we can write the standard point which satisfies the given coordinate system. Now we will write the equation of the circle having radius $10$ and center at origin and substitute the calculated point in the equation and check for the number points that satisfies the condition.

Complete step by step solution:
Given coordinate system is $3x-y-z=0$, $-3x+z=0$, $-3x+2y+z=0$.
Adding the first and second equation we can have
$3x-y-z+\left( -3x+z \right)=0+0$
Removing the parentheses by using mathematical rule that when we multiply a positive sign with a negative sign, we will get negative sign, then we will have
$\begin{aligned} & 3x-y-z-3x+z=0 \\\ & \Rightarrow y=0 \\\ \end{aligned}$
Substituting the above calculated $y$ in the third equation, then we will get
$\begin{aligned} & -3x+2\left( 0 \right)+z=0 \\\ & \Rightarrow -3x+z=0 \\\ \end{aligned}$
From the above equation we can write the value of $z$ as
$z=3x$
Let us assume the value of $x$ as constant say $k$, then the value of $z$ will be
$z=3k$
Hence the point which satisfies the given coordinate system is $\left( x,y,z \right)=\left( k,0,3k \right)$.
Now the equation of the sphere having radius $10$ and center at origin is given by
${{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{10}^{2}}$
If a point lies in the sphere then the point should satisfy the following equation.
${{x}^{2}}+{{y}^{2}}+{{z}^{2}}<100$
Substituting the point $\left( k,0,3k \right)$ in the above expression, then we will get
$\begin{aligned} & {{k}^{2}}+{{0}^{2}}+{{\left( 3k \right)}^{2}}<100 \\\ & \Rightarrow {{k}^{2}}+9{{k}^{2}}<100 \\\ & \Rightarrow 10{{k}^{2}}<100 \\\ \end{aligned}$
Divide the above expression with $10$ on both sides, then we will get
${{k}^{2}}<10$
We can write the values of $k$ which satisfy the above expression are $k=0,\pm 1,\pm 2,\pm 3$.
Hence the number of points which satisfies all the given conditions is $7$.

Note: In this problem we have the simple equations in the coordinate system, so we have simplified the equations in a simple manner. If there are different kinds of equations which cannot be solved in a simple manner, then we need to use any matrix method like crammers’ rule or gauss Jordan method or matrix inverse method to find the solution.