One way is to assume we have 3 variables and 3 equations given by the coefficients on the LHS of the given equations. There is a hidden variable z which is added invisibly in to each equation. The three variables are x, y and z (the hidden variable). This gives a system of three equations from which x, y and z can be found.
(1) 3x+6y+z=12,
(2) 5x+4y+z=19,
(3) x+4y+z=112,
(4)=(2)-(3)=4x=-93, x=-93/4⇒
(5) 4y+z=112-x=112+93/4=541/4=19+465/4=19-5x;
(6)=(2)-(1)=2x-2y=7⇒2y=2x-7=-93/2-7=-107/2, y=-107/4⇒
z=541/4-4y=541/4+107=969/4.
So x=-93/4, y=-107/4, z=969/4. Using these variables we can evaluate the fourth equation:
3+7≡3x+7y+z=-59/4.
This is unlikely to be the right answer because of the awkwardness of -59/4, but it is a possible solution, as derived above.
A more pleasing result includes the order of the equations into the calculations.
This gives whole numbers for the variables and 3+7=-11 when the order of the equations is taken into account.
The interpretation of the equations is:
(1) 3+6≡3x+6y+z+1=12 (n=1)
(2) 5+4≡5x+4y+z+2=19 (n=2)
(3) 1+4≡x+4y+z+3=112 (n=3)
(4) 3+7≡3x+7y+z+4=-11 (n=4), where x=-23, y=-26, z=236, and n is the order of the equations starting at 1.
