Scale either equation (or both) such that the coefficient of one of the variables for both equations will have the same absolute value.
#[1] -4x -3y = 0#
#[2] -x + 4y = 19#
Let's multiply #[1]# by -1 and #[2]# by 4. That way, the coefficient of #x# for both equations will have the same absolute value 4.
#[3] => -1(-4x -3y = 0)#
#[3] => 4x + 3y = 0#
#[4] => 4(-x + 4y = 19)#
#[4] => -4x + 16y = 76#
Now let's add our equations,
#[3] 4x + 3y = 0#
#[4] -4x + 16y = 76#
#[5] 0x + 19y = 76#
#=> y = 4#
Now that we know the value of #y#, let's use that in one of our equations to find #x#. For example, let's use #[2]#
#[2] -x + 4y = 19#
#=> -x + 4(4) = 19#
#=> x = -3#
PS: Note that in this example, we didn't need to multiply #[1]# by -1 in order for #x# to have the same absolute value. If we didn't do that, we'll have to subtract #[1]# and #[4]#. However, I find subtraction error-prone, so I prefer getting rid of subtraction operations and doing addition instead. Besides, even if we didn't do it explicitly, we'll have to negate one of our equations in order to perform subtraction.
