Problems from Problems with polynomials and algebraic fractions

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: $${\displaystyle \frac{7}{13}}={\displaystyle \frac{x}{y}}$$ $$x^2+y^2=5450$$

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:

$$x={\displaystyle \frac{7}{13}}y\Rightarrow ({\displaystyle \frac{7}{13}}y{)}^2+y^2=5450$$

We develop the expression in order to solve for $y$:

$$({\displaystyle \frac{7}{13}}y{)}^2+y^2=5450\iff {\displaystyle \frac{49}{169}}{y}^2+y^2=5450\iff {\displaystyle \frac{218}{169}}{y}^2=5450\iff$$ $$\iff y=\sqrt{{\displaystyle \frac{5450\cdot 169}{218}}}=\sqrt{4225}\iff y=\pm 65$$

We can then obtain $x$: $$x={\displaystyle \frac{7}{13}}y={\displaystyle \frac{7}{13}}\cdot (\pm 65)=\pm 35$$

Solution:

Therefore, the possible equivalent fractions are ${\displaystyle \frac{35}{65}}$ and ${\displaystyle \frac{-35}{-65}}={\displaystyle \frac{35}{65}}$

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