1) x+2y=7, 2x-y=4

Question

What I did:
multiply 2x-y=4 by -2 so I could eliminate the y variable.

-4x-2y = -8
+ x +2y =7
= -3x =-1
Divided -1 by =1 and got x=1/3

I only need someone to check my answer. Also one last question, can I use the elimination method for all system of equations or will there be a time when I absolutely have to use the substitution method?

Answer 1

but go back and get y from either
x+2y=7, 2x-y=4
1/3 + 2 y = 7
2 y = 20/3
y = 10/3
or
2(1/3) -y = 4
y = 2/3 - 4 NOPE so wrong !!!!
I will try it
x+2y=7, 2x-y=4

x + 2 y = 7
-4 x + 2 y = -8
-------------------- subtract
5 x = 15
x = 3 (Not what you had)
back for y
3 +2 y = 7
2 y = 4
y = 2
-------------------------
NOW CHECK THAT
x + 2 y = 7 ?
3 + 4 = 7 YES !
2 x - y = 4 ?
6 - 2 = 4 YES
so I got it right.

Answer 2

error in -4x-2y = -8
you multiplied by -2 so it should have been
-4x+2y = -8 , and you would now have to subtract the two equations

-4x + 2y = -8
x + 2y = 7
-5x = -15
x = 3
you did not find the y
in our case , sub x = 3 into
x+2y = 7
3 + 2y = 7
2y = 4
y = 2

so x = 3, y = 2

Elimination does work all the time, in many cases you will have to perform
some operations on either one or even both equations.

You will have to judge if one method is better than another.
I look at the coefficients of both equations . If I see an x or y term with 1 as coefficient, I might choose substitution.
e.g. in your case one equation is x+2y=7
I would change that to x = 7-2y and sub that into the 2nd
2x-y=4
2(7-2y) - y = 4
14 - 4y - y = 4
-5y = -10
y = 2, then in x = 7-2y , x = 7-4 = 3 , same as above