Problem 66 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

The solutions for the system of nonlinear equations are \(x = ± \sqrt{5.8}\) and \(y = ± \sqrt{4/5}\)

Step 1: Write down the equations

First, present both provided equations: \(x^{2}-y^{2}=5\) and \(3 x^{2}-2 y^{2}=19\)

Step 2: Multiply equations to prepare for addition

To be able to eliminate one variable using addition method, both equations should be written in a way that allows the cancellation of one variable. Multiply the first equation by 2, and the second equation by 1, which yields: \(2x^{2}-2y^{2}=10\) and \(3 x^{2}-2 y^{2}=19\)

Step 3: Add the equations

Now, add the two equations to eliminate \(y^{2}\). The result is \(5x^{2} = 29\), or simplified further \(x^{2} = 29/5\)

Step 4: Solve for x

Take the square root on both sides for solving \(x\). Thus, \(x = \sqrt{29/5} = ± \sqrt{5.8}\)

Step 5: Solve for y

Substitute the obtained values of \(x\) into one of the original equations, the first one for instance. You get \(y^{2} = x^{2} - 5 = 5.8 - 5 = 0.8\). Hence, \(y = \sqrt{0.8} = ± \sqrt{4/5}\)