Question

Resolva o sistema:

{ $\frac{4}{x} - \frac{3}{y} =8$
{ $\frac{5}{x} + \frac{6}{y} =12$

Answer 1

Bom dia

4/x - 3/y = 8
5/x + 6/y = 12 

fazendo X = 1/x e Y = 1/y o sistema vem

4X - 3Y = 8
5X + 6Y = 12
8X - 6Y = 16
13X = 28
X = 28/13 
3Y = 112/13 - 104/13 = 8/13
Y = 8/39 

x = 1/X = 13/28
y = 1/Y = 39/8 

Answer 2

$\textbf{Substitui\c{c}\~ao:}\ \ \mathrm{\boxed{\mathrm{\dfrac{1}{x}=a}}\ \ \mathrm{e}\ \ \boxed{\mathrm{\dfrac{1}{y}=b}}}$

$\begin{cases} \mathrm{\dfrac{4}{x}-\dfrac{3}{y}=8} \\\\ \mathrm{\dfrac{5}{x}+\dfrac{6}{y}=12} \end{cases}\ \to\ \ \ \begin{cases} \mathrm{4a-3b=8\ \mathbf{(I)}} \\ \mathrm{5a+6b=12\ \mathbf{(II)}} \end{cases}$

$\mathbf{Somando\ 2.(I)\ e\ (II):}\\\\ \mathrm{2(4a-3b)+(5a+6b)=2.8+12\ \to\ 8a-6b+5a+6b=16+12\ \to}\\\\ \mathrm{\to\ 13a=28\ \to\ \boxed{\mathbf{a=\dfrac{28}{13}}}}$

$\mathbf{Substituindo\ a\ em\ (I):}\\\\ \mathrm{4a-3b=8\ \to\ b=\dfrac{4}{3}(a-2)=\dfrac{4}{3}\bigg(\dfrac{28}{13}-2\bigg)=\dfrac{4}{3}\bigg(\dfrac{28}{13}-\dfrac{26}{13}\bigg)=}\\\\ \mathrm{=\dfrac{4}{3}\bigg(\dfrac{2}{13}\bigg)\ \to\ \boxed{\mathbf{b=\dfrac{8}{39}}}}$

$\mathbf{Calculando\ x\ e\ y:}\\\\ \mathrm{\dfrac{1}{x}=a\ \to\ \dfrac{1}{x}=\dfrac{28}{13}\ \to\ \boxed{\mathbf{x=\dfrac{13}{28}}}}\\\\\\ \mathrm{\dfrac{1}{y}=b\ \to\ \dfrac{1}{y}=\dfrac{8}{39}\ \to\ \boxed{\mathbf{y=\dfrac{39}{8}}}}$

$\textbf{Solu\c{c}\~ao:}\ \ \boxed{\boxed{\mathbf{(x;y)=\bigg(\dfrac{13}{28};\dfrac{39}{8}\bigg)}}}$