Problems with polynomials and algebraic fractions
Find a fraction equivalent to $\dfrac{7}{13}$ whose squared terms add up to $5450$.
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 $x$: $$x=\dfrac{7}{13}y=\dfrac{7}{13}\cdot(\pm65)=\pm35$$
Therefore, the possible equivalent fractions are $\dfrac{35}{65}$ and $\dfrac{-35}{-65}=\dfrac{35}{65}$