Question

-Consider the following system of equations:--57 - 25How many solutions does the system have?zeroonetwo

Answer 1

Given the system of equations:

$\begin{cases}-\frac{x^2}{3}=-\frac{5}{6}+\frac{y^2}{3}\ldots(1) \\ 5y^2=\frac{25}{2}-5x^2\ldots(2)\end{cases}$

We multiply by 6 the equation (1):

$\begin{gathered} -\frac{6\cdot x^2}{3}=-\frac{6\cdot5}{6}+\frac{6\cdot y^2}{3} \\ \Rightarrow-2x^2=-5+2y^2 \\ \Rightarrow2x^2+2y^2=5\ldots(1^{\prime}) \end{gathered}$

Now, we multiply by 2/5 on the equation (2):

$\begin{gathered} \frac{2}{5}\cdot5y^2=\frac{2}{5}\cdot\frac{25}{2}-\frac{2}{5}\cdot5x^2 \\ \Rightarrow2y^2=5-2x^2 \\ \Rightarrow2x^2+2y^2=5\ldots(2^{\prime}) \end{gathered}$

We can see that both equations (1') and (2') are equal, so we conclude that there are infinitely many solutions.