First, let's multiply the first equation by two on the both sides: <span>8x + 7y = 39 /2 </span>⇒ 16x + 14y = 78
Now, the system is: <span>16x + 14y = 78 </span><span>4x – 14y = –68 </span> 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: <span>8x + 7y = 39 </span>⇒ 8 · 1/2 + 7y = 39 4 + 7y = 39 7y = 39 - 4 7y = 35 ⇒ y = 35 ÷ 7 = 5
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.
Sample Response: Using the linear combination method, you can multiply the first equation by 2 and add the equations to get 20x = 10. Dividing both sides by 20, x = 1/2. To solve for y, substitute 1/2 for x in the equation 8x + 7y = 39 to get 4 + 7y = 39. Solving this equation, y = 5. Checking this in the other equation, 4(1/2) – 14(5) = –68 results in 2 – 70 = –68 or –68 = –68. The solution of the system shown 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. Since 20(1/2) = 10, the system 8x + 7y = 39 and 20x = 10 also has a solution of (1/2, 5).
