x and y

Solve the following system:
{4 x+10 y = 90 | (equation 1)
11 x+5 y = 90 | (equation 2)
Swap equation 1 with equation 2:
{11 x+5 y = 90 | (equation 1)
4 x+10 y = 90 | (equation 2)
Subtract 4/11 × (equation 1) from equation 2:
{11 x+5 y = 90 | (equation 1)
0 x+(90 y)/11 = 630/11 | (equation 2)
Multiply equation 2 by 11/90:
{11 x+5 y = 90 | (equation 1)
0 x+y = 7 | (equation 2)
Subtract 5 × (equation 2) from equation 1:
{11 x+0 y = 55 | (equation 1)
0 x+y = 7 | (equation 2)
Divide equation 1 by 11:
{x+0 y = 5 | (equation 1)
0 x+y = 7 | (equation 2)
Collect results:
Answer: |x = 5                    and                    y = 7

finding x and y, what is 4x+10y and 11x+5y. both are equal to 90.

determinant method:

$\begin{array}{|rcll|} \hline x &=& \frac{ \begin{vmatrix} 90 & 10\\90 & 5\end{vmatrix} } { \begin{vmatrix} 4 & 10\\11 & 5\end{vmatrix} } \\\\ &=& \frac{ 90\cdot \begin{vmatrix} 1 & 10\\1 & 5\end{vmatrix} } { 4\cdot 5 - 11 \cdot 10 } \\\\ &=& \frac{ -90\cdot \begin{vmatrix} 10 & 1\\5 & 1\end{vmatrix} } { -90 } \\\\ &=& \begin{vmatrix} 10 & 1\\5 & 1\end{vmatrix} \\\\ &=& 10-5 \\\\ &=& \mathbf{5} \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline y &=& \frac{ \begin{vmatrix} 4 & 90\\11 & 90\end{vmatrix} } { \begin{vmatrix} 4 & 10\\11 & 5\end{vmatrix} } \\\\ &=& \frac{ 90\cdot \begin{vmatrix} 4 & 1\\11 & 1\end{vmatrix} } { 4\cdot 5 - 11 \cdot 10 } \\\\ &=& \frac{ -90\cdot \begin{vmatrix} 1 & 4\\1 & 11\end{vmatrix} } { -90 } \\\\ &=& \begin{vmatrix} 1 & 4\\1 & 11\end{vmatrix} \\\\ &=& 11-4 \\\\ &=& \mathbf{7} \\ \hline \end{array}$