Simultaneous equation by elimination when no pair of variable has equal coefficient but with same sign.

1. Solve simultaneous linear equations by Elimination
Method
By Hector Castellanos

2. No Equal coefficients and same
sign
Given two equations: 2x +3y = 12
 4x +2y =16
Note that the coefficients are not equal for any set of
variables.

3. Label Equations
Let 2x + 3y = 12 be equation 1
 4x +2y =16 be equation 2
The objective is to get a set of variables to have equal
coefficients and opposite in sign.

4. Since eq 1 is 2x + 3y = 12
 and eq 2 is 4x +2y =16
A common multiple for 3 and 2 for the variable y
would be 6. Therefore:
 Multipliy eq 1 by 2
 2(2x +3y = 12)
 4x +6y =24 eq 3
 Multiply eq 2 by 3
 3(4x +2y =16)
 12x + 6y =48 eq 4
The objective is to get a set of variables to have equal
coefficients and then subtract(opposite in sign) an
equation from the other to eliminate a variable.

5. Solve for x
Subtract eq 4 from eq 3 having now equal coefficients
and same sign to eliminate a variable.
 4x + 6y =24 eq 3
 -(12x + 6y =48) eq 4
 4x-12x + 6y-6y = 24-48
 -8x = -24
 -8x
/-8= -24
/-8 ÷ both sides by -8 to get x itself
 x = 3
Since x=3 then substitute into eq 1 or 2.

6. Solve for y
Since x = 3 and eq 1 is 2x + 3y = 12
 2x + 3y =12 eq 1
 2(3) + 3y =12
 6 + 3y =12
 6-6 + 3y = 12-6 subtract 6 to both sides
 3y = 6
 3y
/3 = 6
/3 ÷ both sides by 3
 y = 2
So x=3 and y= 2

7. Prove
Since x = 3 and y= 2
 2x + 3y = 12 and 4x +2y = 16
2(3) + 3(2) = 12 and 4(3) + 2(2) =16
 6 + 6 = 12 and 12 + 4 = 16
So it holds true for both equations.

8. Conclusion
Label both equations (Example eq 1 and eq 2).
Multiple equations to get a common coefficient for a
set of variable
Subtract one equation from the other to eliminate the
said variable.
Substitute for the solved variable and solve for the
eliminated variable.