Simultaneous Equations The Elimination method If a pair

Simultaneous Equations The Elimination method

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)

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

2. 3 x + 4 y = 11 (i) -3 x + 2 y = 1 (ii) Solve The x-terms in both equations are the same, but the 3 x – 3 x. So+we 4 yadd + 2 y 11 equations +1 signs are different. the=two to eliminate x. 6 y = 12 Then put this value in equation (i) y=2 3 x + 4 y = 11 3 x + (4 x 2) = 11 3 x + 8 = 11 3 x = 3 x = 1 (i)

SUBTRACTING THE TWO EQUATIONS (i) 1. Solve 2 x + y = 7 (ii) x + y= 4 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. 2 x – x + y – y = 7 – 4 Then put this value in x=3 equation (i) 2 x + y = 7 (2 x 3) + y = 7 6 +y y = 7 = 1 (i)

2. Solve 3 x + 4 y = 11 (i) 3 x + 2 y = 7 (ii) The x-terms in both equations are the same and the 3 xthe – 3 x + 4 y. So– we 2 y subtract = 11 – 7 signs are also SAME. the two equations to eliminate x. 2 y = 4 y=2 3 x + 4 y = 11 3 x + (4 x 2) = 11 3 x + 8 = 11 3 x = 3 x = 1 (i)

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.

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.

1. Solve 2 x + 3 y = 13 x + 2 y = 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). 2 x + 4 y = 16 2 x + 3 y = 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. 2 x – 2 x + 4 y – 3 y = 16 – 13 y=3 2 x + 3 y = 13 2 x + (3 x 3) = 13 2 x + 9 = 13 2 x = 4 x = 2 (i)