Find a fraction equivalent to $$\dfrac{7}{13}$$ whose squared terms add up to $$5450$$.
Development:
First, we must identify the unknowns and give them a name. In our case, for example, x might be the numerator of the fraction that we are looking for, and y the denominator. We then have the system that we must solve: $$$\dfrac{7}{13}=\dfrac{x}{y}$$$ $$$x^2+y^2=5450$$$
The system consists of a polynomial and an algebraic fraction. We will isolate a variable of the algebraic fraction and will replace it in the second equation:
$$$x=\dfrac{7}{13}y \Rightarrow \Big(\dfrac{7}{13}y\Big)^2+y^2=5450$$$
We develop the expression in order to solve for $$y$$:
$$$\Big(\dfrac{7}{13}y\Big)^2+y^2=5450 \Leftrightarrow \dfrac{49}{169}y^2+y^2=5450 \Leftrightarrow \dfrac{218}{169}y^2=5450 \Leftrightarrow$$$ $$$\Leftrightarrow y=\sqrt{\dfrac{5450\cdot169}{218}}=\sqrt{4225} \Leftrightarrow y=\pm65$$$
We can then obtain $$x$$: $$$x=\dfrac{7}{13}y=\dfrac{7}{13}\cdot(\pm65)=\pm35$$$
Solution:
Therefore, the possible equivalent fractions are $$\dfrac{35}{65}$$ and $$\dfrac{-35}{-65}=\dfrac{35}{65}$$