Problem 57 - College Algebra

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 are \(x = \sqrt[]{\frac{29}{5}}\) and \(y = \pm\sqrt[]{(\sqrt[]{\frac{29}{5}})^2 - 5}\).

Step 1: Identify the coefficients of the terms

Examine both equations in your system. The coefficients are the numbers in front of your variables. In this case, for the first equation, the coefficient of \(x^{2}\) is 1 and the coefficient of \(y^{2}\) is -1. For the second equation, the coefficient of \(x^{2}\) is 3 and the coefficient of \(y^{2}\) is -2.

Step 2: Multiply equations to make coefficients cancel

To apply the addition method, coefficients of one of the variable terms must be the same but with different signs. In our case, for instance, we can manipulate the two equations to make coefficients of \( y^{2}\) equal to eliminate the variable. To do this, multiply equation 1: \(x^{2}-y^{2}=5\), by 2 and equation 2: \(3 x^{2}-2 y^{2}=19\), by 1. The equations become:2. \(2x^{2}-2y^{2}=10\)and3. \(3x^{2}-2y^{2}=19\).

Step 3: Add the equations

Now, add equation 2 and equation 3 together. The sum is: \(2x^{2} + 3x^{2} = 10 + 19\), resulting in: \(5x^{2} = 29\).

Step 4: Solve for x

Now we have a simple equation and can solve it for \(x\). To do this, divide both sides by 5, then take the square root. Hence, \(x = \sqrt[]{\frac{29}{5}}\).

Step 5: Substitute x in original equation to solve for y

Now substitute \(x = \sqrt[]{\frac{29}{5}}\) into the first original equation: \(x^{2}-y^{2}=5\). By simplifying, we can solve for \(y\) which results in \(y = \pm \sqrt[]{(\sqrt[]{\frac{29}{5}})^2 - 5}\).