| Example Solve
Solution: By multiplying by (L. C. M. of 7 and 8) 56 , we get 8 x - 7 y = 56 . . . (1) 7 x - 8 y = 34 . . . (2) Adding (1) and (2)
8 x - 7 y = 56 7 x - 8 y = 34 15 x -15 y = 90 x - y = 6 . . . (3)
Subtracting (2) from (1) 8 x - 7 y = 56 7 x - 8 y = 34 (-) (+) (-) x + y = 22 . . . (4) Adding (3) and (4) x - y = 6 x + y = 22 2 x = 28 x = 14 Subtracting (3) and (4) x - y = 6 x + y = 22 (-) (-) (-) - 2 y = -16 y = 8
Therefore, x = 14, y = 8 is the required solution. Note 1 Consider x - y = 8. This equation is satisfied by x = 10, y = 2, x = 1, y = 3; x = 12, y = 4 and so on. But if there are two such equations (of the first degree in x and y ) simultaneously holding good, then we have in general only one solution. Note 2 We stated ’in general’ in the last note; because the two equations may be of this type: x - y = 8; 4 x - 4 y = 32. In this case the two equations are not independent. Equation (2) can be obtained from (1) by multiplying both sides by 4. The two equations are therefore really equivalent to a single equation Again consider two equations, 2 x + y = 3 . . . (1) 8 x + 4 y = 4 . . . (2) These equations are "inconsistent" i.e. they can not hold simultaneously. For if we multiply each side of (1) by 4, we have 8 x + 4 y = 12 but (2) shows that 8 x + 4 y = 13 which implies that 12 = 13, which is totally absurd. On solving these two by the usual process we have 0 = 1; which is absurd.
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7.1 Definition |
