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

Multiplying the first equation by 2 and adding the equations results in 20x = 10. The solution of the system is (1/2, 5). The system 8x + 7y = 39 and 20x = 10 is formed by replacing 4x –14y = –68 by a sum of it and a multiple of 8x + 7y = 39.

Answer 2

First, let's multiply the first equation by two on the both sides:
8x + 7y = 39 /2
⇒ 16x + 14y = 78

Now, the system is:
16x + 14y = 78
4x – 14y = –68

After adding this up in the column:
(16x + 4x) + (14y - 14y) = 78 - 68
20x = 10
⇒ x = 10/20 = 1/2


y can be calculated by replacin the x:
8x + 7y = 39
⇒ 8 · 1/2 + 7y = 39
4 + 7y = 39
7y = 39 - 4
7y = 35
⇒ y = 35 ÷ 7 = 5