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Chapter 13: Problem 66

Explain how to solve a nonlinear system using the addition method. Use \(x^{2}-y^{2}=5\) and \(3 x^{2}-2 y^{2}=19\) to illustrate your explanation.

Short Answer

Expert verified
The solution set to the system of equations \(x^{2}-y^{2}=5\) and \(3x^{2}-2y^{2}=19\) is obtained by first eliminating one variable, then solving for the remaining variable and substituting back to find the possible values for the eliminated variable. This results in a set of possible (x, y) pairs that satisfy both equations simultaneously.

Step by step solution

01

Elimination of a Variable

To eliminate one variable, the equations are multiplied by certain factors so that the resulting equations will cancel out one variable when added together. Here, if the second equation is multiplied by \(\frac{1}{2}\), then the modified system of equations will be that: \(x^{2}-y^{2}=5\) and \(\frac{3}{2}x^{2}-y^{2}=\frac{19}{2}\).
02

Adding the Equations Together

Next, add the two equations together. This will result in \(\left(x^{2} + \frac{3}{2}x^{2}\right) - \left(y^{2}+y^{2}\right) = \left(5 + \frac{19}{2}\right)\), which simplifies to \(\frac{5}{2}x^{2} - 2y^{2} = \frac{29}{2}\).
03

Solving for the Remaining Variable

After simplification, the equation: \(\frac{5}{2}x^{2} - 2y^{2} = \frac{29}{2}\) remains. The equation can further be simplified to: \(5x^{2} - 4y^{2} = 29\). Solving for \(x\) in terms of \(y\) gives the equation: \(x = \sqrt{\frac{29+4y^{2}}{5}}\).
04

Substitution

Substitute the expression \(x = \sqrt{\frac{29+4y^{2}}{5}}\) back into one of the original equations (e.g. \(x^{2}-y^{2}=5\)) to solve for \(y\).
05

Finding the Solution Set

Solving for \(y\) in the original equation with the substituted \(x-value\) gives two possible values for \(y\). Then substitute these \(y-values\) into the expression for \(x\) to get two corresponding \(x-values\). This will give the set of ordered pairs that are solutions to the system of equations.

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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. $$x=y^{2}+4 y$$

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