Problem 94 - College Algebra with Modeling and Visualization

Question

Use elimination to solve the nonlinear system of equations. $$ \begin{array}{r} x^{2}+2 y=15 \\ 2 x^{2}-y=10 \end{array} $$

Step-by-Step Solution

Answer

The solutions are \((\sqrt{7}, 4)\) and \((-\sqrt{7}, 4)\).

Step 1: Set Up System of Equations

We are given two equations: 1. \( x^2 + 2y = 15 \) 2. \( 2x^2 - y = 10 \). The goal is to eliminate one variable so that we can solve for the other.

Step 2: Align and Multiply Equations

First, align the system of equations so we have a common coefficient for one of the variables. Multiply equation (1) by 1 (essentially keep it the same) and equation (2) by 2, to make the coefficients of \( y \) the same (but opposite in sign): \( 2(x^2 + 2y) = 2(15) \) which simplifies to:- \( x^2 + 2y = 15 \)- \( 4x^2 - 2y = 20 \)

Step 3: Add the Equations to Eliminate y

Add the modified equations together to eliminate \( y \):\[(x^2 + 2y) + (4x^2 - 2y) = 15 + 20\]Simplifying this gives:- \( x^2 + 4x^2 = 35 \)- \( 5x^2 = 35 \)

Step 4: Solve for x

Divide both sides by 5:\[x^2 = 7\]Take the square root of both sides:\( x = \pm \sqrt{7} \)

Step 5: Substitute x Back into an Original Equation

Substitute \( x = \pm \sqrt{7} \) into the first original equation to solve for \( y \): For \( x = \sqrt{7} \):- Plug into \( x^2 + 2y = 15 \) which becomes:- \( 7 + 2y = 15 \)- \( 2y = 8 \)- \( y = 4 \)For \( x = -\sqrt{7} \):- The above process will yield the same value for \( y \).

Step 6: Write the Solution

The solutions to the system of equations are \((x, y) = (\sqrt{7}, 4)\) and \((x, y) = (-\sqrt{7}, 4)\). Both pairs satisfy both original equations.

Key Concepts

Nonlinear Systems

Nonlinear systems of equations involve at least one equation that is not linear, meaning at least one equation does not describe a straight line when graphed. In the exercise given, both equations involve terms that have exponents greater than one, making this a nonlinear system because of the presence of the term \(x^2\). Nonlinear systems can be more complex than linear systems because they may have more solutions or no solutions at all.
To solve these systems, various methods such as substitution, elimination, or graphical methods can be used. The elimination method works by eliminating one of the variables. This transforms the nonlinear system into a simpler form.
Solutions to nonlinear systems can represent points, curves, or surfaces, depending on the nature of each equation. In this exercise, solving for \(x\) first allows us to better understand the behavior of the system as a whole, leading to a full solution.

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. In mathematics, solving a system of equations means finding all values of the variables that satisfy all equations simultaneously.
There are different types of systems:

• Consistent: Has one or more solutions.
• Inconsistent: Has no solution.
• Dependent: Equations that describe the same line or curve, having infinitely many solutions.

An equation within a system can be linear or nonlinear. Linear equations form straight lines when graphed, while nonlinear equations can form a variety of shapes, like hyperbolas or circles.
In the given exercise, we have a nonlinear system due to the quadratic terms. By employing the elimination method, we remove one variable to simplify the problem, allowing us to find the solutions that satisfy all parts of the system efficiently.

Mathematics Education

In mathematics education, understanding how to solve systems of equations, whether linear or nonlinear, is a base skill that builds towards more advanced math topics. The elimination method is a fundamental technique that helps students grasp how to manipulate equations to isolate variables.
This exercise highlights the importance of practicing different methods of solving equations. It develops critical thinking skills and enhances understanding of algebraic concepts.
Learning through step-by-step examples, like the one provided, students see the logical progression of solving a problem.

• Step-by-step solutions help break down complex problems into manageable parts.
• Students learn to identify key terms and variables.
• It enhances skills in recognizing patterns and applying mathematical operations effectively.

This step-by-step approach aids in fostering a deeper understanding of mathematical principles, preparing students for further education in mathematics and related fields.