# To calculate: The solution for the system of equations 5x^{2}+y^{2}

To calculate: The solution for the system of equations $5{x}^{2}+{y}^{2}=14$ and ${x}^{2}-2{y}^{2}=-17$. If the system does not have a unique solution, determine whether the system is inconsistent, or the equations are dependent.
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May Dunn

Calculation:
Consider the provided system of equations:
$5{x}^{2}+{y}^{2}=14$.......(1)
And,
${x}^{2}-2{y}^{2}=-17$.....(2)
Multiply by 2 in equation (1) $10{x}^{2}+2{y}^{2}=28$......(3)
$11{x}^{2}=11$
${x}^{2}=1$
$x=±1$
Now, substitute 1 for xin equation (2) and obtain the value of y:
$\left(1{\right)}^{2}-2{y}^{2}=-17$
$1-2{y}^{2}=-17$
$-2{y}^{2}=-18$
${y}^{2}=9$
$y=±3$
Again, put $x=$ and $y=-3$ in equation $5{x}^{2}+{y}^{2}=14$:
$5{\left(1\right)}^{2}+{\left(-3\right)}^{2}\stackrel{?}{=}14$
$5+9\stackrel{?}{=}14$
$14\stackrel{?}{=}14$
Put $x=1$ and $y=—3$ in equation ${x}^{2}-2{y}^{2}=-17$
${\left(1\right)}^{2}-2{\left(-3\right)}^{2}\stackrel{?}{=}-17$
$1-18\stackrel{?}{=}-17$
$-17\stackrel{?}{=}-17$
Again, put $x=-1$ and $y=3$ in equation $5{x}^{2}+{y}^{2}=14$
$5{\left(-1\right)}^{2}+{\left(3\right)}^{2}\stackrel{?}{=}14$
$5+9\stackrel{?}{=}14$
$14\stackrel{?}{=}14$
Put $x=-1$ and $y=3$ in equation ${x}^{2}-2{y}^{2}=-17$
${\left(-1\right)}^{2}-2{\left(3\right)}^{2}\stackrel{?}{=}-17$
$1-18\stackrel{?}{=}-17$
$-17\stackrel{?}{=}-17$
Again, put $x=-1$ and $y=-3$ in equation $5{x}^{2}+{y}^{2}=14$
$5{\left(-1\right)}^{2}+{\left(-3\right)}^{2}\stackrel{?}{=}14$
$5+9\stackrel{?}{=}14$

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