Correct answer: infinitely many solutions
Explanation:
The equation x - 4y = 5 is a linear equation in two variables.
A linear equation in two variables represents a straight line on the coordinate plane and has infinitely many solutions, since there are infinitely many points (x, y) that satisfy the equation.
Therefore, the given equation has infinitely many solutions.
For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form:
\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \).
Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let
\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).
If the given figure shows the graph of polynomial \( y = ax^2 + bx + c \), then: