Problem 56 - College Algebra

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 of the non-linear system are (2/5, -7/5) and (2, 1)

Step 1: Isolate one variable in one equation

From the given system of equations, take the linear equation \(2x - y = 3\) and isolate \(y\). When \(y\) is isolated, the equation becomes \(y = 2x - 3 \)

Step 2: Substitute the isolated variable into the second equation

Substitute \(y\) from the first equation into the second equation \(x^2 + y^2 = 9\). This would give \[x^2 + (2x - 3)^2 = 9\].

Step 3: Solve the equation

Expand, simplify, and factor out the resulting equation, \(x^2 + (2x - 3)^2 = 9\), to get \(5x^2 -12x+4=0\). Solve for \(x\), which gives two possible values, \(x=2/5\) or \(x=2\).

Step 4: Substitute the values of \(x\) into equation 1 to get corresponding y-values

Substitute \(x = 2/5\) into \(y = 2x - 3 \) to obtain \(y = -7/5\). Alternatively, for \(x = 2\), the corresponding \(y\) becomes \(y = 1\). Therefore, the solutions of the given non-linear system of equations are \((2/5, -7/5)\) and \((2, 1)\)