Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{l} -x^{2}+y=2 \\ -4 x+y=-1 \end{array} $$

Step-by-Step Solution

Answer

The solutions are \((1, 3)\) and \((3, 11)\).

Step 1: Express y from the second equation

The second equation is \[-4x + y = -1\]. To express \( y \) in terms of \( x \), add \( 4x \) to both sides. So, \[ y = 4x - 1 \].

Step 2: Substitute expression for y into the first equation

Take the expression for \( y \) from Step 1 and substitute it into the first equation \[-x^2 + y = 2\]. Replacing \( y \), we get \[-x^2 + (4x - 1) = 2\].

Step 3: Simplify and solve for x

Simplify the equation from Step 2: \[-x^2 + 4x - 1 = 2\]. Add 1 to both sides: \[-x^2 + 4x = 3\]. Multiply through by -1 to get: \[x^2 - 4x = -3\]. Re-arrange this to get a standard quadratic form: \[x^2 - 4x + 3 = 0\]. Factor this equation: \[(x-1)(x-3) = 0\]. Thus, \( x = 1 \) or \( x = 3 \).

Step 4: Solve for y using x values

Use the expression for \( y \) from Step 1, \( y = 4x - 1 \). First, substitute \( x = 1 \): \[y = 4(1) - 1 = 3\]. Then, substitute \( x = 3 \): \[y = 4(3) - 1 = 11\]. So, the solution pairs are \((1, 3)\) and \((3, 11)\).

Key Concepts

Solving Nonlinear Equations

Nonlinear equations are those where the variables have power greater than one, are multiplied together, or involve functions like exponents, logarithms, and trigonometric identities. Unlike linear equations, which graph as straight lines, nonlinear equations can create curves, making them a bit trickier to solve.
To tackle nonlinear systems, we can use several methods. Some include substitution, elimination, or graphical methods. Each method has its own benefits depending on the situation. For instance, substitution is particularly handy when one of the equations is already solved for one variable, as seen in this exercise.
Solving these systems often requires strategic algebraic manipulations to simplify complex equations into a form that is easier to analyze and solve.

Substitution Method

The substitution method is a powerful tool for solving systems of equations, especially when dealing with a mix of linear and nonlinear equations. The aim is to express one variable in terms of the other.
Once a variable is expressed in terms of the other, this expression can be substituted back into the remaining equations. This is clearly demonstrated in the exercise where the second equation was used to express \( y = 4x - 1 \) and then substitute this into the first nonlinear equation.
When using substitution, keep the following in mind:

• Ensure the expression is simplified as much as possible before substitution.
• Check for any extraneous solutions that may arise after substitution, especially when dealing with nonlinear equations.
• Always verify the solution by substituting back into the original equations.

Quadratic Equations

Quadratic equations are a specific type of nonlinear equations where the highest exponent of the variable is two. The general form is \( ax^2 + bx + c = 0 \). In this exercise, we isolated a quadratic equation by substituting values and rearranging terms.
Once in standard form, quadratic equations can be solved by factoring, completing the square, or using the quadratic formula. In our example, the equation was successfully factored into \((x-1)(x-3) = 0\), allowing us to find solutions of \( x = 1 \) and \( x = 3 \).
Key elements to remember about quadratic equations:

• Always bring the equation to a standard form before solving.
• Check for all possibilities of factoring; sometimes recognizing patterns can help.
• Consider other methods like the quadratic formula if factoring isn't straightforward.

Quadratic equations are a fundamental component in mathematics, frequently appearing in different types of nonlinear problems.