What is the solution to the system of equations?

What is the solution to the system of equations y = 2x + 3 and y = x + 6

Use substitution, so you get x+6 = 2x+3. This then becomes 6 = x+3
so x=3. Plug this in for either equation and you'll get y=9.

For the second system.
3x + y = 5
2x + 3y = 8

Solve for x first:
|5 1|
|8 3|
-------
|3 1|
|2 3|

The determinants of those matrices are (15-8) / (9-2) = 7/7 = 1
The solution for x = 1. To solve for y, you can use Cramer's rule again or plug into one of the two equations.
Cramer's rule:
|3 5|
|2 8|
------
7

This comes out to (24-10)/7 = 14/7 = 2.
Thus the solution for y = 2.

You can plug either of these into the system and get the expected answers.