answer:The solution is given by x=bmu+anv. This satisfies the first congruence because bmuequiv bpmod{m}, and it satisfies the second congruence because anvequiv apmod{n}.

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answer:1. To rewrite the equation in standard form, we complete the square for the y terms: 7 x+3 y^2-4 y-10=0 3 y^2-4 y=10-7 x 3 (y^2-frac{4}{3}y)=10-7 x 3 (y^2-2cdotfrac{2}{3}y+(frac{2}{3})^2)=10-7 x+(frac{2}{3})^2 3 (y-frac{2}{3})^2=10-7 x+frac{4}{9} 3 (y-frac{2}{3})^2=frac{94}{9}-7 x 2. The equation is now in standard form for a parabola: ay^2+bxy+cx+dy+e=0 where a=3, b=0, c=-7, d=0, and e=frac{94}{9}. 3. Since a=3>0, the parabola opens upward. 4. The vertex of the parabola is given by the formula: V=(frac{-b}{2a}, frac{-d}{2a}) V=(frac{-0}{2cdot3}, frac{-0}{2cdot3}) V=(frac{0}{6}, frac{0}{6}) V=(0, 0) 5. The directrix of the parabola is given by the equation: x=V_x-frac{1}{4a} x=0-frac{1}{4cdot3} x=-frac{1}{12} 6. The focal parameter of the parabola is given by the formula: p=frac{1}{4a} p=frac{1}{4cdot3} p=frac{1}{12} 7. The focus of the parabola is given by the formula: F=(V_x+p, V_y) F=(0+frac{1}{12}, 0) F=(frac{1}{12}, 0) Classification: Parabola Equation: 3 y^2-4 y=10-7 x Vertex: left(frac{7}{6}, frac{2}{3}right) Directrix: x=frac{185}{84} Focal Parameter: frac{7}{6} Focus: left(frac{29}{28}, frac{2}{3}right)

answer:The Moment Generating Function is a function of one variable, t. It represents the expected value of e^{tX}, where t is a scalar and X is a random variable. The use of M_X(t) indicates that the function depends on the distribution of X through the expectation, but it is not a function of the random variable X itself. Similarly, E[g(X)] is a function of X because g depends on X, but in the case of the MGF, e^{tX} is a function of both t and X, yet the expectation operation yields a function of t only, due to the linearity of expectation. Thus, M_X(t) is not comparable to f(x,y), which represents a joint probability density function, while f_X(y) is a marginal probability density function. The focus of the MGF is on the behavior of the distribution as t varies, not on the individual realizations of the random variable X.