# To find: The exact location of the ship tracked by

To find: The exact location of the ship tracked by the tracking system.
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Procedure used:
An Application of a Nonlinear System:
"Step 1: Write a system of equations modeling the conditions in the problem.
Step 2: Solve the system and answer the question asked in the problem.
Step 3: Check the proposed solution in the original wording of the problem."
Calculation:
It is given that the ship lies on a path described by $2{y}^{2}-{x}^{2}=1$ but the boat located on a different path is described by $2{x}^{2}-{y}^{2}=1$ when the process is repeated.
Use the above procedure to find the exact location of the ship.
Multiply $2{y}^{2}-{x}^{2}=1$ by 2 and obtain the equation $4{y}^{2}-2{x}^{2}=2$.
Add the both equations and obtain the result as follows.
$\left(4{y}^{2}-2{x}^{2}\right)+\left(2{x}^{2}-{y}^{2}\right)=2+1$
$\left(4{y}^{2}-{y}^{2}\right)+\left(2{x}^{2}-2{x}^{2}\right)=3$
$3{y}^{2}=3$
${y}^{2}=1$
$y=±1$.
Substitute $y=1$ in the equation $2{x}^{2}-{y}^{2}=1$ and obtain the value of x as follows.
$2{x}^{2}-{\left(1\right)}^{2}=1$
$2{x}^{2}-1=1$
${x}^{2}=\frac{1+1}{2}$
${x}^{2}=1$
$x=±1$
Substitute $y=-1$ in the equation $2{x}^{2}-{y}^{2}=1$ and obtain the value of x as follows.
$2{x}^{2}-{\left(-1\right)}^{2}=1$
$2{x}^{2}-1=1$
${x}^{2}=\frac{1+1}{2}$
${x}^{2}=1$
$x=±1$.
Thus, for $y=1$, $x=±1$ and for $y=-1,x=±1$.
Thus, the solution set is $\left\{\left(1,1\right),\left(-1,1\right),\left(1,-1\right),\left(-1,-1\right)\right\}$
However, the ship is located in the first quadrant of the coordinate system.
Hence, the point is (1,1).
Check the result, by substituting the obtained solutions in the given original equations $2{y}^{2}-{x}^{2}=1$ and $2{x}^{2}-{y}^{2}=1$.
Substitute (1,1) in the given system and check.
$2{\left(1\right)}^{2}-{\left(1\right)}^{2}=1$
$1=1$
$2{\left(1\right)}^{2}-{\left(1\right)}^{2}=1$
$1=1$
Therefore, the exact location of the ship is (1,1).