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Chapter 5: Problem 57

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

Step by step solution

01

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.
02

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. \(3x^{2}-2y^{2}=19\).
03

Add the equations

Now, add equation 2 and equation 3 together. The sum is: \(2x^{2} + 3x^{2} = 10 + 19\), resulting in: \(5x^{2} = 29\).
04

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, \(x = \sqrt[]{\frac{29}{5}}\).
05

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

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Most popular questions from this chapter

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(8,5)\\\ &5 x-4 y=20\\\ &3 y=2 x+1 \end{aligned} $$

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two cquations?

In Exercises \(19-30,\) solve each system by the addition method. \(3 x=4 y+1\) \(3 y=1-4 x\)

In Exercises \(19-30,\) solve each system by the addition method. \(2 x+3 y=-16\) \(5 x-10 y=30\)

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(4 x-2 y=2\) \(2 x-y=1\)
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