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Chapter 13: Problem 65

Explain how to solve a nonlinear system using the substitution method. Use \(x^{2}+y^{2}=9\) and \(2 x-y=3\) to illustrate your explanation.

Short Answer

Expert verified
The solution to the system of equations \(x^{2}+y^{2}=9\) and \(2 x-y=3\) is \(x = 1, y = -1\) and \(x = -2, y = -7\).

Step by step solution

01

Solve the linear equation for one variable

To begin, solve the linear equation \(2 x-y=3\) for y, which gives \(y = 2x - 3\).
02

Substitute this solution into the nonlinear equation

Substitute \(y = 2x - 3\) into the first equation. That gives us a quadratic equation in terms of x, \(x^{2}+(2x-3)^{2}=9\).
03

Solve the quadratic equation

By solving this quadratic equation, derive the roots as \(x = 1\) and \(x= -2\).
04

Find the corresponding y values

Now substitute these x values into the equation from step 1, \(y = 2x - 3\). With \(x = 1\), we get \(y = -1\) and with \(x = -2\), we get \(y = -7\).
05

Check the solutions

Finally, back-substitute the pairs \(x = 1, y = -1\) and \(x = -2, y = -7\) into both original equations to ensure they are true.

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