Simultaneous Equations: One linear and One non-liear

Note on simultaneous equations, one linear and one non-linear

1. Simultaneous Equations One Linear and One Quadratic Solve the equations: x 2 – y 2 = 7 and x + y = 7 Solution x 2 – y 2 = 7 _____ ( i ) x + y = 7 _____ (ii) From ( i ) ( x + y ) ( x – y ) = 7 7 ( x – y ) = 7 x – y = 1 _____ (iii) (ii) + (iii) x + y = 7 + x – y = 1 2 x = 8 x = 4 Substitute x in (ii) 4 + y = 7 y = 7 – 4 = 3

2. Simultaneous Equations Miscellaneous Problems Solve the equation: 3 x + 5 y = 2 x – 4 y + 13 = 5 x + y – 4. Solution 8 x + 2 2 x + 1 = 2 x + 3 (2 3 ) x + 2 2 x 3 + n 2 – 4 n = 0 n 2 – 4 n + 3 = 0 ( n ) ( n ) – 3 – 1 = 0 n = 3 or n = 1 log 2 x = 3 or log 2 x = 1 x = 2 3 or x = 2 1 x = 8 or x = 2 × 2 1 = 2 x × 2 3 (2 x ) 3 + (2 x ) 2 × 2 = 2 x × 8 Let 2 x = m m 3 + 2 m 2 = 8 m m 3 + 2 m 2 – 8 m = 0 m ( m 2 + 2 m – 8) = 0 m ≠ 0 but m 2 + 2 m – 8 = 0