Solving Nonlinear Systems Equations Step-by-Step Guide

Jul 15, 2025 by ADMIN 55 views

Solving Nonlinear Systems of Equations: A Comprehensive Guide

This article provides a detailed guide on solving nonlinear systems of equations, focusing on the specific examples provided. We will explore the methods for finding solutions and discuss the number of solutions possible for each system. Understanding how to solve these systems is crucial in various fields, including mathematics, physics, and engineering. Nonlinear systems, unlike linear systems, can have curves and other complex shapes, leading to multiple solutions or no solutions at all. This makes the solution process more intricate and requires a deeper understanding of algebraic and graphical techniques. Let's dive into the methods and examples to illustrate the process clearly.

Question 14: Solving the System

To address the first system of equations, which includes a parabola and a line, we can utilize the substitution method. This approach involves setting the two equations equal to each other, creating a single equation in one variable, which can then be solved using algebraic techniques. The first equation is a quadratic equation representing a parabola, while the second is a linear equation representing a straight line. The solutions to the system represent the points where the line and parabola intersect. These points satisfy both equations simultaneously. By finding these points, we determine the common solutions that lie on both curves. The process involves careful algebraic manipulation and a thorough understanding of quadratic equations. Understanding the nature of parabolas and lines is crucial for interpreting the solutions.

The given system is:

\begin{align*} y &= x^2 - 4x + 4 \ y &= 2x - 5 \end{align*}

Step 1: Substitution Method

Since both equations are solved for y, we can set them equal to each other:

$x^2 - 4x + 4 = 2x - 5$
Step 2: Simplify the Equation

Move all terms to one side to set the equation to zero:

$x^2 - 4x + 4 - 2x + 5 = 0$

$x^2 - 6x + 9 = 0$
Step 3: Solve the Quadratic Equation

The resulting quadratic equation can be factored:

$(x - 3)(x - 3) = 0$

This simplifies to:

$(x - 3)^2 = 0$

So, $x = 3$
Step 4: Find the Value of y

Substitute $x = 3$ into either of the original equations. We'll use the second equation:

$y = 2(3) - 5$

$y = 6 - 5$

$y = 1$
Step 5: State the Solution

The solution to the system is the point $(3, 1)$ . This means the parabola and the line intersect at only one point. Therefore, there is one solution.

Detailed Explanation of the Solution Process

In this question, the core task is to find the point(s) where the parabola defined by $y = x^2 - 4x + 4$ intersects the line defined by $y = 2x - 5$ . The substitution method is particularly effective here because it allows us to eliminate one variable and solve for the other. By setting the expressions for y equal to each other, we transform the problem into solving a quadratic equation. The key step is recognizing that the resulting quadratic equation, $x^2 - 6x + 9 = 0$ , is a perfect square, which simplifies the factoring process. Factoring the equation as $(x - 3)^2 = 0$ reveals that there is only one solution for x, which is $x = 3$ . This indicates that the line is tangent to the parabola, touching it at only one point. Substituting this value back into the linear equation gives the corresponding y-value, $y = 1$ . Thus, the solution (3, 1) represents the single point of intersection between the parabola and the line.

Visual Representation and Geometric Interpretation

A graphical representation of the equations further clarifies the solution. The parabola $y = x^2 - 4x + 4$ is a U-shaped curve, and the line $y = 2x - 5$ is a straight line. When plotted on the same coordinate plane, it becomes evident that the line touches the parabola at only one point, confirming the single solution we found algebraically. This tangency implies that the line has the same slope as the parabola at the point of intersection, which is a unique characteristic of this solution. Understanding the geometric interpretation helps in visualizing the solution and verifying the algebraic results. The graph not only confirms the number of solutions but also provides a visual context for the algebraic manipulations performed.

Importance of Checking the Solution

It is always a good practice to check the solution by substituting the values of x and y back into both original equations to ensure they are satisfied. In this case, substituting $x = 3$ and $y = 1$ into both equations confirms that these values satisfy both equations, thereby validating the solution. This step is crucial in preventing errors and ensuring the accuracy of the results. Checking the solution is particularly important in nonlinear systems, where errors can easily occur during the algebraic manipulations. The process of verification reinforces the correctness of the solution and builds confidence in the answer.

Question 15: Solving the System

Moving on to the second system of equations, we apply a similar approach using the substitution method to find the solutions. This system also involves a quadratic equation and a linear equation, but the specific forms of the equations will lead to a different outcome in terms of the number of solutions. The techniques for solving remain the same, but the algebraic steps and the interpretation of the results may vary. Understanding the differences in the equations and how they affect the solutions is a key aspect of solving nonlinear systems. The interplay between the coefficients and constants in the equations determines the number and nature of the intersections.

The given system is:

\begin{align*} y &= x^2 - 8x + 15 \ y &= 2x - 6 \end{align*}

Step 1: Substitution Method

Set the two equations equal to each other:

$x^2 - 8x + 15 = 2x - 6$
Step 2: Simplify the Equation

Move all terms to one side:

$x^2 - 8x + 15 - 2x + 6 = 0$

$x^2 - 10x + 21 = 0$
Step 3: Solve the Quadratic Equation

Factor the quadratic equation:

$(x - 3)(x - 7) = 0$

This gives two solutions for x:

$x = 3$ or $x = 7$
Step 4: Find the Values of y

Substitute $x = 3$ and $x = 7$ into the second equation:

For $x = 3$ :

$y = 2(3) - 6$

$y = 6 - 6$

$y = 0$

For $x = 7$ :

$y = 2(7) - 6$

$y = 14 - 6$

$y = 8$
Step 5: State the Solutions

The solutions to the system are the points $(3, 0)$ and $(7, 8)$ . This indicates that the parabola and the line intersect at two distinct points. Therefore, there are two solutions.

Comprehensive Analysis of the Two Solutions

In this question, the system of equations consists of a parabola $y = x^2 - 8x + 15$ and a line $y = 2x - 6$ . By employing the substitution method, we equate the expressions for y, resulting in the quadratic equation $x^2 - 10x + 21 = 0$ . Unlike the previous question, this quadratic equation has two distinct real roots. Factoring the equation as $(x - 3)(x - 7) = 0$ yields two solutions for x: $x = 3$ and $x = 7$ . This signifies that the line intersects the parabola at two different points. Substituting each x-value back into the linear equation allows us to find the corresponding y-values. For $x = 3$ , we find $y = 0$ , and for $x = 7$ , we find $y = 8$ . Thus, the solutions to the system are the points (3, 0) and (7, 8). These points represent the coordinates where the line and the parabola intersect, and they satisfy both equations simultaneously.

Graphical Representation and Intersection Points

A graphical representation of the parabola and the line provides a visual confirmation of the two solutions. Plotting the parabola $y = x^2 - 8x + 15$ and the line $y = 2x - 6$ on the same coordinate plane reveals that they intersect at two distinct points. These intersection points correspond to the solutions we found algebraically: (3, 0) and (7, 8). The graph illustrates that the line crosses the parabola at two locations, indicating that there are two pairs of x and y values that satisfy both equations. This visual representation enhances the understanding of the solutions and provides a concrete interpretation of the algebraic results. The graph not only confirms the number of solutions but also offers insights into the geometric relationship between the parabola and the line.

Verifying the Solutions and Ensuring Accuracy

As with any mathematical problem, verifying the solutions is a crucial step to ensure accuracy. Substituting the values of x and y for each solution back into the original equations confirms that they satisfy both equations. For the solution (3, 0), substituting $x = 3$ and $y = 0$ into both equations confirms their validity. Similarly, substituting $x = 7$ and $y = 8$ into both equations verifies the second solution. This process of verification is particularly important in nonlinear systems, where the complexity of the equations can lead to errors. By checking the solutions, we can be confident in the correctness of our answers. This step reinforces the understanding of the solutions and their significance in the context of the given system of equations.

Conclusion

In summary, solving nonlinear systems of equations often involves using methods like substitution to reduce the system to a single equation in one variable. The number of solutions depends on the nature of the equations and their graphical representations. Understanding these techniques is fundamental in mathematics and its applications. The two examples we discussed demonstrate how a system of a quadratic equation and a linear equation can have one or two solutions, depending on the specific equations. The solutions represent the points of intersection between the curves, providing a geometric interpretation of the algebraic results. Mastering these methods allows for the accurate and efficient resolution of nonlinear systems, which are prevalent in various scientific and engineering contexts. The ability to solve these systems is a valuable skill in both theoretical and practical applications.

Nonlinear Systems of Equations, Substitution Method, Quadratic Equations, Linear Equations, Number of Solutions, Parabola, Intersection Points, Algebraic Techniques, Graphical Representation, Verification of Solutions.