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Chapter 13: Problem 47

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 2 x^{2}+x y=6 \\ x^{2}+2 x y=0 \end{array}\right.$$

Short Answer

Expert verified
The system of equations has two solutions: \((-2\sqrt{3}, \sqrt{3})\) and \((2\sqrt{3}, -\sqrt{3})\).

Step by step solution

01

Simplify one equation for one variable

Any of the two equations can be arranged for either \(x\) or \(y\). Let's arrange the second equation for \(x\): Respectively, divide the second equation by \(x\), we get: \(x + 2y = 0\) or \(x = -2y\).
02

Substitute the obtained value in the other equation

Substitute \(x = -2y\) from the first step into the first equation: \(2(-2y)^{2} + (-2y)y = 6\), which simplifies to: \(4y^{2} - 2y^{2} = 6\) or \(2y^{2} = 6\).
03

Solve for \(y\)

Dividing by \(2\) on both sides of the equation gives: \(y^{2} = 3\). Taking the square root on both sides gives: \(y = \sqrt{3}\) and \(y = -\sqrt{3}\). There are two values of \(y\) as the square root of a number yields a positive and a negative value.
04

Substitute \(y\) back into one of the original equations to solve for \(x\)

Substitute \(y = \sqrt{3}\) into the equation \(x = -2y\): \(x = -2\sqrt{3}\). And also substitute \(y = -\sqrt{3}\) into the equation \(x = -2y\): \(x = 2\sqrt{3}\). Hence the solutions are \((-2\sqrt{3}, \sqrt{3})\) and \((2\sqrt{3}, -\sqrt{3})\).

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Most popular questions from this chapter

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y=-x^{2}-4 x+4$$

If you are given an equation of a parabola, explain how to determine if the parabola opens to the right, to the left, upward, or downward.

How can you distinguish ellipses from hyperbolas by looking at their equations?

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. $$x=y^{2}-6 y+8$$

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. $$y=2(x-3)^{2}+1$$
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