a)
[tex]\left\{ {{x + 2y = 3} \atop {2x + my = 7}} \right. \\\left \{ {{x = 3 - 2y} \atop {2(3-2y) + my = 7}} \right. \\\left \{ {{x = 3 - 2y} \atop {6 - 4y + my = 7}} \right. \\\left \{ {{x = 3 - 2y} \atop {6 - 4y + my = 7}} \right. \\\left \{ {{x = 3 - 2y} \atop {(m-4)y = 1}} \right. \\\left \{ {{x = 3 - 2y} \atop {y = \frac{1}{m-4} }} \right. \\\left \{ {{x = 3 - \frac{2}{m-4}} \atop {y = \frac{1}{m-4} }} \right.\\\\[/tex]
x > 0
3 - 2/(m-4) > 0 ; m ≠ 4
2/(m-4) < 3
2(m-4) < 3(m-4)²
2m - 8 < 3(m² - 8m + 16)
2m - 8 < 3m² - 24m + 48
-3m² + 26m - 56 < 0
Δ = 4 ; √Δ = 2
m₁ = -26-2 / -6 = -28/-6 = 14/3
m₂ = -26+2 / -6 = -24/-6 = 4
m ∈ (-∞, 4)u(14/3, +∞)
y > 0
1 / (m-4) > 0 ; m≠ 4
m-4 > 0
m > 4
m ∈ (4, +∞)
m ∈ (14/3, +∞)
b)
[tex]\left\{ {{x+y=m} \atop {2x-y=m}} \right. \\\left\{ {{x=m-y} \atop {2(m-y)-y=m}} \right. \\\left\{ {{x=m-y} \atop {2m-2y-y=m}} \right. \\\left\{ {{x=m-y} \atop {-3y=-m}} \right. \\\left\{ {{x=m-y} \atop {y=\frac{m}{3} }} \right.\\\left\{ {{x=m-\frac{m}{3}} \atop {y=\frac{m}{3} }} \right.\\\left\{ {{x=\frac{2m}{3}} \atop {y=\frac{m}{3} }} \right.[/tex]
x > y
2m/3 > m/3
2m > m
m > 0
m ∈ (0, +∞)
