You'll learn better techniques for higher order systems, but the 2nd order system (2 eqtns with 2 unknowns) is straightforward to solve by hand. Here are the general steps

1) Place both equations in the form aX + bY = c

2) Inspect the constants of both equations to find a GCM for either X or Y

3) Multiply Either equation (or both) to achieve the GCM in both eqtns for that X or Y

4) Add or subtract the equations to get RID of that X or Y

5) Solve for that remaining Y or X

6) Substitute that first answer (YorX) to compute the other (XorY)

Let's do your example

1)

x-y=3

2x-y=5

2)Looking at the above, you'll note that if we subtract the 1st equation from the 2nd, we can make the Y-variable disappear. To be clear, I'll rewrite them

2x-y=5

-( x-y)=3

x = (5-3)=2

6) Now just pick an equation to substitute back to find y (doesn't matter which one you use)

-y=3-x (moved x to the right on 1st eqtn)

y=x-3 (multiplied by -1 on both sides)

y=(2)-3= -1 (substituted 2 into x from above)-----> (x,y)=(2,-1)

6) Lets check the other eqtn just to verify

-y =-2x + 5

y= 2x - 5 (multiplied by -1 on both sides)

y=2(2)-5= -1 (substituted 2 into x from above)-----> (x,y)=(2,-1) VERIFIED!

## Comments