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Chapter 13: Problem 21

Solve each system by the addition method. $$\left\\{\begin{array}{l} x^{2}-4 y^{2}=-7 \\ 3 x^{2}+y^{2}=31 \end{array}\right.$$

Short Answer

Expert verified
The solutions to the system of equations are \[x = \pm \sqrt{\frac{38}{5}}\] and \[y = \pm \sqrt{\frac{24}{5}}\].

Step by step solution

01

Rewrite equations and take the difference

Rewrite both the equations as \(x^2 = 4y^2 - 7\) and \(3x^2 = 31 - y^2\). Now, take the difference of two equations (equation of \(3x^2\) - equation of \(x^2\)) which gives \(2x^2 = y^2 + 7\).
02

Find \(y^2\) from the difference equation

Rearrange the equation into the form \(y^2\). The equation becomes \(y^2 = 2x^2 - 7\).
03

Substitute \(y^2\) in the original system of equations

Substitute \(y^2\) from step 2 into the \(x^2\) term in the second equation of the original system. This results in the equation, \(3x^2 = 31 - (2x^2 - 7)\). Simplifying gives \(5x^2 = 38\). Thus, \[x = \pm \sqrt{\frac{38}{5}}\].
04

Solve for y

Insert the found x values into the equation of \(y^2\). It yields \(y^2 = 2(\pm \sqrt{\frac{38}{5}})^2 - 7 = 2(\frac{38}{5}) - 7 = \frac{38}{5}*2 - 7 = 24/5\). So, \[ y = \pm \sqrt{\frac{24}{5}}\].

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Most popular questions from this chapter

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. $$x=-3(y-5)^{2}+3$$

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. $$x=(y-3)^{2}-5$$

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. $$x=-(y-5)^{2}+4$$

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. $$y=-(x+1)^{2}+4$$

This will help you prepare for the material covered in the first section of the next chapter. Evaluate \(n^{2}+1\) for all consecutive integers from 1 to 6 Then find the sum of the six evaluations.
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