How to find major & minor axes of the ellipse $10x^2+14xy+10y^2-7=0$? 
What are major & minor axes of the ellipse: $10x^2+14xy+10y^2-7=0$ ? 

My trial:
from given equation:  $10x^2+14xy+10y^2-7=0$
$$10x^2+14xy+10y^2=7$$
$$\frac{x^2}{7/10}+\frac{xy}{7/14}+\frac{y^2}{7/10}=1$$
I know the standard form of ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
But the term of $xy$ is a bottleneck. 
From here I can't proceed. 
Can somebody please help me solve this problem? 
Thank you. 

Yet another method is with matrices and linear algebra. If we stuff $(x,y,1)^T$ into a vector, then the matrix
$${\bf M}=\left[\begin{array}{ccc}10&7&0\\7&10&0\\0&0&-7\end{array}\right]$$
Can be used to express the ellipse as a scalar product with matrix multiplication: 
$$\left[\begin{array}{ccc}x&y&1\end{array}\right] \left[\begin{array}{ccc}10&7&0\\7&10&0\\0&0&-7\end{array}\right] \left[\begin{array}{c}x\\y\\1\end{array}\right]$$
A linear coordinate change can be expressed like this:
$$\left[\begin{array}{c}x_1\\y_1\\1\end{array}\right]= {\bf T}_{0\to 1}\left[\begin{array}{c}x_0\\y_0\\1\end{array}\right]$$
By making an eigenvalue decomposition of $\bf M$ we can find the coordinate system to transform to:
$${\bf M} = \left[\begin{array}{ccc}0&-1/\sqrt 2&1/\sqrt 2\\0&1/\sqrt 2&1/\sqrt 2\\1&0&0\end{array}\right]\left[\begin{array}{ccc}-7&0&0\\0&3&0\\0&0&17\end{array}\right]\left[\begin{array}{ccc}0&-1/\sqrt 2&1/\sqrt 2\\0&1/\sqrt 2&1/\sqrt 2\\1&0&0\end{array}\right]^{-1}$$
Here the coefficients of the new expression appears in the diagonal: $-7,3, 17$. The rest of the steps are similar to @user45914123 answer. 

The main benefit of this method is that we can bake translation into the base change. For example if you had $X^2+2X+1+Y^2=R^2$, you would not know what to do with the $2X$ if you could only do linear transformations on the $[X,Y]^T$ vector.
