Question

Find a fraction equivalent to 5/7 whose squared terms add up to 1184.

Answer 1

The system of equations of two unknowns is formulated and solved.

$\large\displaystyle\text{ \begin{gathered}\sf \bf{ \left\{\begin{matrix} \ \ \ \dfrac{x}{y} = \dfrac{5}{7} \\ x^2+y^2 = 1184 \end{matrix}\right. \ \Longrightarrow \ x=\dfrac{5}{7}y } \end{gathered} }$

$\large\displaystyle\text{ \begin{gathered}\sf \bf{ \left (\dfrac{5}{7}y \right )^2+y^2=1184\ \Longrightarrow\ 25y^2+49y^2=58016 } \end{gathered} }$

                                                              $\large\displaystyle\text{ \begin{gathered}\sf \bf{74y^{2}=58016} \end{gathered} }\\\large\displaystyle\text{ \begin{gathered}\sf \bf{ \ \ \ \ \ \ y^{2}=784 } \end{gathered} }\\\large\displaystyle\text{ \begin{gathered}\sf \bf{ \ \ \ \ \ y=\pm\sqrt{784}=\pm28 } \end{gathered} }$

                                                                  $\large\displaystyle\text{ \begin{gathered}\sf \bf{ x=\dfrac{5}{7}(\pm 28)=\pm 20 } \end{gathered} }$

The fraction that satisfies the request is $\bf{\dfrac{20}{28}}$ , since in $\bf{\dfrac{-20}{-28}}$ the negative signs are canceled and the first fraction is obtained.