1. Simultaneous Equations
The Elimination method

2. If a pair of simultaneous equations contain an x -term which are exactly the same, we can solve them by elimination.
When the signs of the equal terms are DIFFERENT, we ADD together the two equations to eliminate x.
The same method can be used if the y – terms are equal. (Or for that matter any letter term)

3. ADDING THE TWO EQUATIONS
1. Solve 3x + y = 15
2x – y = 5
(ii)
The y-terms in both equations are the same, but the signs are different. So we add the two equations to eliminate y.
3x + 2x + y – y =
5x = 20
x = 4
Then put this value in
equation (i)
3x + y = (i)
(3 x 4) + y = 15
12 + y = 15
y = 3

4. (i) (ii) 3x – 3x + 4y + 2y = 11 + 1 3x + 4y = 11 (i)
2. Solve 3x + 4y = 11
-3x + 2y = 1
(i)
(ii)
The x-terms in both equations are the same, but the signs are different. So we add the two equations to eliminate x.
3x – 3x + 4y + 2y =
6y = 12
y = 2
Then put this value in
equation (i)
3x + 4y = (i)
3x + (4 x 2) = 11
3x = 11
3x = 3
x = 1

5. SUBTRACTING THE TWO EQUATIONS
1. Solve 2x + y = 7
x + y = 4
(ii)
The y-terms in both equations are the same and the signs are also the SAME. So we subtract the two equations to eliminate y.
2x – x + y – y = 7 – 4
x = 3
Then put this value in
equation (i)
2x + y = 7 (i)
(2 x 3) + y = 7
6 + y = 7
y = 1

6. (i) (ii) 3x – 3x + 4y – 2y = 11 – 7 3x + 4y = 11 (i)
2. Solve 3x + 4y = 11
3x + 2y = 7
(i)
(ii)
The x-terms in both equations are the same and the signs are also the SAME. So we subtract the two equations to eliminate x.
3x – 3x + 4y – 2y = 11 – 7
2y = 4
y = 2
3x + 4y = (i)
3x + (4 x 2) = 11
3x = 11
3x = 3
x = 1

7. Simultaneous Equations
If the equal terms in both equations are the same, but the signs are different.
We add the two equations to eliminate one unknown.
If the equal terms in both equations are the same and the signs are also the SAME.
We subtract the two equations to eliminate one unknown.

8. Simultaneous Equations
If neither the x-term or y-term are the same the elimination method will not work.
However, it is possible to form a pair of equations by multiplying one or both equations by a number.
NB. The multiplication must be applied to all parts of the equation.

9. (i) (ii) 2x – 2x + 4y – 3y = 16 – 13 2x + 3y = 13 (i)
1. Solve 2x + 3y = 13
x + 2y = 8
(i)
(ii)
Neither x or y terms are the same. But if I multiply equation (ii) by 2 an equal term can be created with equation (i).
2x + 4y = 16
2x + 3y = 13
The x-terms in both equations are the same and the signs are also the SAME. So we subtract the two equations to eliminate x.
2x – 2x + 4y – 3y = 16 – 13
y = 3
2x + 3y = (i)
2x + (3 x 3) = 13
2x = 13
2x = 4
x = 2