Problem 65 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

The solutions are \(x = 0, y = -3\) and \(x = 12 / 5, y = 3 / 5\).

Step 1: Isolate a variable

Rearrange the second equation \(2 x-y=3\) to isolate y. Add y to both sides and subtract 3 from both sides to get \(y = 2x - 3\).

Step 2: Substitution

Substitute \(y = 2x - 3\) in the first equation \(x^{2}+y^{2}=9\). This gives: \(x^{2}+(2x - 3)^{2}=9\).

Step 3: Simplify and Solve for x

Simplify to get a quadratic equation: \(x^{2} + 4x^{2} - 12x + 9 = 9\). Combine similar terms and subtract 9 from both sides to get \(5x^{2} - 12x = 0\). Factor out \(x\) to get \(x(5x - 12) = 0\). Setting each factor equal to 0 gives the solutions \(x = 0\) or \(x = 12 / 5\).

Step 4: Solve for y

Substitute \(x = 0\) and \(x = 12 / 5\) in the equation \(y = 2x - 3\) to get the corresponding y values. For \(x = 0\), \(y = -3\), and for \(x = 12 / 5\), \(y = 3 / 5\).

Step 5: Solution

Hence the solutions to the system of equations are \(x = 0, y = -3\) and \(x = 12 / 5, y = 3 / 5\).