Question

Show and explain how replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions as the one shown.

8x + 7y = 39

4x – 14y = –68

Answer 1

Elementary row/column operations: replacing one equation by that equation plus a multiple of the other does not change the solution set. Reason: if (x,y) satisfies both original equations E1 and E2, then it also satisfies E1 + k·E2 for any k. Conversely, if (x,y) satisfies E2 and E1' = E1 + k·E2, then subtracting k·E2 from E1' recovers E1, so (x,y) satisfies the original system. Thus the two systems are equivalent.

Apply this to the given system:
E1: 8x + 7y = 39
E2: 4x − 14y = −68

Replace E1 by E1 − 2·E2 (so k = −2):
E1 − 2·E2 = (8x + 7y) − 2(4x − 14y) = 8x + 7y − 8x + 28y = 35y,
right-hand side: 39 − 2(−68) = 39 + 136 = 175.
So the new system is
35y = 175
4x − 14y = −68

From 35y = 175 we get y = 5. Substitute into E2:
4x − 14(5) = −68 → 4x − 70 = −68 → 4x = 2 → x = 1/2.

Check: 8(1/2) + 7(5) = 4 + 35 = 39 and 4(1/2) − 14(5) = 2 − 70 = −68. Thus (x,y) = (1/2, 5) is the solution, and the replacement step preserved the solution set.