I have tried the addition method on this problem where I multiple both equations if the constant but my teacher swears there is another better way so please help me

3x + 7y= -1
2x + 3y= 6
_______________
6X + 14Y = -2
-6X - 9Y = -18
________________

5Y = -20
Y = -4
________________
2x + 3y = 6
2X -12 = 6
2X = 6+12
2X = 18
X =9

_______________________

X=9
Y = -4
___________________________
proof:-

3x + 7y= -1 ****27-28 = -1
2x + 3y= 6 *****18-12 = 6

You need to solve one of the equations for x or y and substitute into the other one. It will be a little messy. Fractions.

Well, strictly speaking, yes, using substitution will give rise to fractions, making it a little messy.

You can try substituting something else instead of x or y to make is easier.

3x + 7y= -1
2x + 3y= 6

Multiply (I) by 2 and (ii) by 3.

6x + 14y= -2
6x + 9y= 24

Make 6x the subject of formula in the first equation.

6x = -2 - 14y

Now, substitute 6x by (-2 -14y) into the other equation, giving:

(-2 - 14y) + 9y= 24

You substituted 6x instead of x. Now it's easier.

Why didn't I think of that? (Can't spread rep).

Don't worry :) This happens to me as well.