Question

em que x .y ≠ 0 . Nesse caso, marque a alternativa que apresenta o valor equivalente para a expressão 3x - y

Answer 1

$\displaystyle \left \{ {{\displaystyle \frac{3}{\text x}-\frac{6}{\text y}=\frac{1}{6}} \atop {\displaystyle \frac{2}{\text x}+\frac{3}{\text y}=\frac{1}{2}}}$

Vamos tirar o MMC das duas equações  :

$\displaystyle \underline{\text{1{\ª} equa{\c c}{\~a}o}}:\\\\ \frac{3\text y-6\text x}{\text {xy}}=\frac{1}{6} \\\\ 6(3\text y-6\text x ) = \text{xy} \\\\ \boxed{\text {xy} = 18\text y-36\text x } \\\\\\ \underline{\text{2{\ª} equa{\c c}{\~a}o}}: \\\\ \frac{2\text y+3\text x}{\text{xy}}=\frac{1}{2} \\\\ \boxed{\text{xy}=4\text y+6\text x }$

Igualando :

$\displaystyle 18 \text y -36\text x = 4\text y+6\text x \\\\ 14\text y = 42\text x \\\\ \boxed{\text y = 3\text x}$

Substituindo na 2ª equação :

$\displaystyle \frac{2}{\text x}+\frac{3}{\text y}=\frac{1}{2} \\\\\ \text{xy} =4\text y + 6\text x \\\\ \text x.3\text x= 4.3\text x + 6\text x\\\\ 3\text x^2=12\text x+ 6\text x \\\\ 3\text x^2 = 18\text x \\\\ \text x^2-6\text x = 0 \\\\ \text x \neq 0 \ ; \ \boxed{\text x =6}$

Achando o valor de y  :

$\text y = 3\text x \\\\ \text y = 3.6 \\\\ \boxed{\text y = 18}$

A questão pede 3x - y , ou seja :

$3\text x - \text y = 3.6-18 \\\\ 3\text x-\text y = 18-18 \\\\\ \huge\boxed{\ \text{3x}-\text y =0\ }\checkmark$