4x + 5y = 812
x + y = 186

Solve using gaussian elimination:

|   4   5   |   812 |
|   1   1   |   186 |

R1 --> 1/4R1

|   1   5/4 |   203 |
|   1   1   |   186 |

R2 --> -R1 + R2
R2 --> -R2

|   1   5/4 |   203 |
|   0   1/4 |   17  |

R2 --> 4R2

|   1   5/4 |   203 |
|   0   1   |   68  |

R1 --> -5/4R2 + R1

|   1   0   |   118 |
|   0   1   |   68  |

So...

x = 118
y = 68