Question

Solve the system. Use any method you wish.

$\left\{\begin{array}{l}\frac{2}{x^2}-\frac{3}{y^2}+1=0\\ \frac{6}{x^2}-\frac{7}{y^2}+2=0\end{array}\right.$

Short answer

The solution of system of equations are $\left(2,-\sqrt{2}\right),\left(2,\sqrt{2}\right),\left(-2,-\sqrt{2}\right),\left(-2,\sqrt{2}\right)$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\left\{\begin{array}{l}\frac{2}{x^2}-\frac{3}{y^2}+1=0......\left(i\right)\\ \frac{6}{x^2}-\frac{7}{y^2}+2=0.......\left(ii\right)\end{array}\right.$

Multiply equation (i) by $x^2{y}^2$,

$$\begin{gathered} \frac{2x^2{y}^2}{x^2}-\frac{3x^2{y}^2}{y^2}+x^2{y}^2=0 \\ 2y^2-3x^2+x^2{y}^2=0......\left(iii\right) \end{gathered}$$

Multiply equation (ii) by $x^2{y}^2$,

$$\begin{gathered} \frac{6x^2{y}^2}{x^2}-\frac{7x^2{y}^2}{y^2}+2x^2{y}^2=0 \\ 6y^2-7x^2+2x^2{y}^2=0......\left(iv\right) \end{gathered}$$

Step 2. Multiply equation (iii) by 3.

Multiply equation (iii) by $3$,

$6y^2-9x^2+3x^2{y}^2=0......\left(v\right)$

Subtract equation (v) and (iv),

$$\begin{gathered} \left(6y^2-9x^2+3x^2{y}^2\right)-\left(6y^2-7x^2+2x^2{y}^2\right)=0 \\ 6y^2-9x^2+3x^2{y}^2-6y^2+7x^2-2x^2{y}^2=0 \\ 2x^2=x^2{y}^2 \\ y=-\sqrt{2},\sqrt{2} \end{gathered}$$

Step 3. Substitute the values of y in equation (iv).

Substitute $y=\sqrt{2}$in equation (iv),

$$\begin{gathered} 6{\left(\sqrt{2}\right)}^2-7x^2+2x^2{\left(\sqrt{2}\right)}^2=0 \\ 12-7x^2+4x^2=0 \\ 12=3x^2 \\ x=2,-2 \end{gathered}$$

Substitute $y=-\sqrt{2}$in equation (iv),

$$\begin{gathered} 6{\left(-\sqrt{2}\right)}^2-7x^2+2x^2{\left(-\sqrt{2}\right)}^2=0 \\ 12-7x^2+4x^2=0 \\ 12=3x^2 \\ x=2,-2 \end{gathered}$$

Therefore, the solution sets are $\left(2,-\sqrt{2}\right),\left(2,\sqrt{2}\right),\left(-2,-\sqrt{2}\right),\left(-2,\sqrt{2}\right)$.