OK, let's do one all the way from the beginning:

10x + y = 35 and 4x - 7y = -23

We've got to rewrite one of these equations so it's like

y = x stuff     or    x = y stuff

And here's the key to making your life happier:

You don't want fractions!

If possible, you want to pick a guy to solve for that doesn't create fractions.

Check it out:

If we solve for this guy...                    it's nice and clean!

10x + y = 35 ... y = 35 - 10x ... circle it and stick it into the OTHER equation ... 4x - 7y = -23

4x - 7 ( 35 - 10x ) = -23 ... solve for x ... 4x - 245 + 70x = -23 ... 74x - 245 = -23 ... 74x = 222 ... x = 3 ... Whew!  That was getting icky!

Now, you stick this guy

 

into circled guy!

x = 3 ... stick it into y = 35 - 10x ... y = 35 - 10 ( 3 ) ... y = 5

I think we've got it...  (3, 5)...  I'll let you check it!  * In both equations!