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Chapter 13: Problem 23

Solve each system by the addition method. $$\left\\{\begin{array}{l} 3 x^{2}+4 y^{2}-16=0 \\ 2 x^{2}-3 y^{2}-5=0 \end{array}\right.$$

Short Answer

Expert verified
The system of equations has four solutions: (2, 1), (2, -1), (-2, sqrt(5/3)) and (-2, -sqrt(5/3)).

Step by step solution

01

Multiply the Equations to Make the Coefficient of y-squared the Same

Begin by multiplying the first equation by 3 and the second by 4. This results in: \[9x^{2}+12y^{2}-48=0\] [and] \[8x^{2}-12y^{2}-20=0\] respectively. This makes the coefficients on \(y^{2}\) the same in each equation, which will be important for the next step.
02

Use the Addition Method

The addition method is a technique where both equations are combined in a way that cancels one of the variables to solve for the other variable. Here, adding the new first and second equations will provide: \[17x^{2}-68=0.\]
03

Solve for the Variable x

Work out the value of x by solving \[17x^{2}-68=0.\] We get two solutions: \[x=\sqrt{4}\] (equivalent to x=2) or \[x=-\sqrt{4}\] (equivalent to x=-2)
04

Substitute x into One of the Original Equations

Take each value for x and substitute it back into either of the two original equations to calculate y. From this, the two pairs of (x, y) solutions for the system of equations are found. When x=2 we get \[y=卤\sqrt{1}\] or y=卤1. When x=-2 we get \[y=卤\sqrt{5/3}\].

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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. $$x=-(y-3)^{2}+4$$

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. $$x-7-8 y=y^{2}$$

Indicate whether the graph of each equation is a circle, an ellipse, a hyperbola, or a parabola. $$x-3-4 y=6 y^{2}$$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=-3(y-1)^{2}-2$$

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