Solve the system of equations: 2x² + y² = 1, 6 + 7x² + |x| + 7x = 1

The system of equations is:
1) 2x² + y² = 1
2) 6 + 7x² + |x| + 7x = 1
From (1), y² = 1 - 2x². Since y² ≥ 0, we have 1 - 2x² ≥ 0, which implies x² ≤ 1/2, or -1/√2 ≤ x ≤ 1/√2.
Substitute y² into (2) is not possible directly. Let's simplify (2): 7x² + |x| + 7x + 5 = 0.
Case 1: x ≥ 0. Then |x| = x. The equation becomes 7x² + x + 7x + 5 = 0, so 7x² + 8x + 5 = 0. The discriminant is Δ = 8² - 4(7)(5) = 64 - 140 = -76 < 0. There are no real solutions for x ≥ 0.
Case 2: x < 0. Then |x| = -x. The equation becomes 7x² - x + 7x + 5 = 0, so 7x² + 6x + 5 = 0. The discriminant is Δ = 6² - 4(7)(5) = 36 - 140 = -104 < 0. There are no real solutions for x < 0.
Since there are no real solutions for x from the second equation, the system has no real solutions.