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Chapter 13: Problem 51

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} \frac{3}{x^{2}}+\frac{1}{y^{2}}=7 \\ \frac{5}{x^{2}}-\frac{2}{y^{2}}=-3 \end{array}\right.$$

Short Answer

Expert verified
The solution to the system of equations could contain multiple pairs of \(x\) and \(y\) values, depending on the solutions of the quadratic equation for \(y\). Each possible \(y\) value would have a corresponding \(x\) value found by substituting the \(y\) value back into the \(x^{2}\) expression.

Step by step solution

01

Express the First Equation in terms of \(x\)

Let's express the first equation in terms of \(x\). Multiply the entire equation by \(x^{2}y^{2}\) to get rid of the denominators. This gives us: \(3y^{2}+x^{2}=7x^{2}y^{2}\). Now we can express \(x^{2}\) as: \(x^{2}=\frac{3y^{2}}{7y^{2}-1}\)
02

Substitute \(x^{2}\) into the Second Equation

Now we substitute the expression we found for \(x^{2}\) into the second equation and simplify. This gives us: \(5*\frac{3y^{2}}{7y^{2}-1}-\frac{2}{y^{2}}=-3\). Now it's a matter of solving this equation for \(y\). Multiply the entire equation by \((7y^{2}-1)y^{2}\) to get rid of the denominators, giving us: \(15y^{4}-5y^{2}-2(7y^{2}-1)=-3(7y^{2}-1)y^{2}\). Next, simplify the equation and solve for \(y\) by using the Quadratic Formula.
03

Find the Corresponding \(x\) Values

Once the \(y\) values have been calculated, substitute them back into the \(x^{2}\) expression to find the corresponding \(x\) values: \(x^{2}=\frac{3y^{2}}{7y^{2}-1}\). Use square root to find \(x\).

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Most popular questions from this chapter

The equation of a parabola is given. Determine: a. if the parabola is horizontal or vertical. b. the way the parabola opens. c. the vertex. $$x=-y^{2}+4 y+1$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-2 y^{2}-4 y$$

Use the vertex and intercepts to sketch the graph of each equation. If needed, find additional points on the parabola by choosing values of y on each side of the axis of symmetry. $$x=-2 y^{2}+4 y-3$$

This will help you prepare for the material covered in the first section of the next chapter. Find the product of all positive integers from \(n\) down through 1 for \(n=5\)

Graph: \(3 x-2 y \leq 6\)
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