Solving Systems Of Equations A Step-by-Step Guide

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In this article, we will delve into the process of finding solutions for a system of equations, specifically focusing on the problem David is trying to solve. The system of equations presented is:

-4x - 7 = y
x² - 2x - 6 = y

We will explore the different methods to determine the solutions and analyze the nature of these solutions – whether they are real numbers, unique, or non-existent. Our main goal is to provide a clear and comprehensive explanation to help you understand how to solve such systems effectively. This exploration will not only address the specific problem at hand but also equip you with the knowledge to tackle similar challenges in the future.

Understanding the Problem and Possible Solutions

When dealing with a system of equations, our primary objective is to find the values of the variables that satisfy all equations simultaneously. In this case, we have two equations with two variables, x and y. The solutions we seek are the pairs of values (x, y) that make both equations true. The given options suggest different scenarios:

  • No real number solutions: This implies that there are no values of x and y that can satisfy both equations. Graphically, this would mean that the two equations do not intersect.
  • One unique real number solution: This indicates that there is exactly one pair of values (x, y) that satisfies both equations. Graphically, this corresponds to the two equations intersecting at a single point.
  • Multiple real number solutions: This suggests that there are several pairs of values (x, y) that satisfy both equations. Graphically, this could mean that the two equations intersect at multiple points or that they represent the same line.

Our task is to determine which of these scenarios is true for the given system of equations. To do this, we will employ algebraic methods to solve the system and analyze the resulting solutions.

Solving the System of Equations A Detailed Walkthrough

To solve the system of equations,

-4x - 7 = y
x² - 2x - 6 = y

we can use the substitution method. Since both equations are expressed in terms of y, we can set them equal to each other:

-4x - 7 = x² - 2x - 6

This gives us a quadratic equation in terms of x. To solve this, we first need to rearrange the equation into the standard quadratic form, which is ax² + bx + c = 0. Adding 4x and 7 to both sides of the equation, we get:

0 = x² - 2x - 6 + 4x + 7

Simplifying the equation, we have:

0 = x² + 2x + 1

Now we have a quadratic equation in the standard form. We can solve this equation using several methods, such as factoring, completing the square, or the quadratic formula. In this case, the equation can be easily factored.

Factoring the Quadratic Equation

We are looking for two numbers that multiply to 1 and add up to 2. The numbers 1 and 1 satisfy these conditions. Therefore, we can factor the quadratic equation as:

(x + 1)(x + 1) = 0

This can also be written as:

(x + 1)² = 0

Finding the Value of x

To find the value of x, we set the factor equal to zero:

x + 1 = 0

Solving for x, we get:

x = -1

So, we have one unique value for x, which is -1.

Finding the Value of y

Now that we have the value of x, we can substitute it back into either of the original equations to find the value of y. Let's use the first equation:

y = -4x - 7

Substituting x = -1, we get:

y = -4(-1) - 7
y = 4 - 7
y = -3

Thus, the value of y is -3.

Analyzing the Solution and Determining the Correct Statement

We have found one unique solution for the system of equations, which is (x, y) = (-1, -3). This means that there is one pair of real numbers that satisfies both equations. Now, let's compare our solution with the given statements:

  • A. There are no real number solutions. This statement is incorrect because we found a real number solution.
  • B. There is one unique real number solution at (-1, -3). This statement is correct because we found exactly one solution, and it is (-1, -3).
  • C. There are two real number solutions. This statement is incorrect because we found only one solution.

Therefore, the correct statement is B. There is one unique real number solution at (-1, -3).

Graphical Interpretation of the Solution

To further understand the solution, we can consider the graphical representation of the equations. The first equation, y = -4x - 7, represents a straight line with a slope of -4 and a y-intercept of -7. The second equation, y = x² - 2x - 6, represents a parabola.

The solution to the system of equations is the point where the line and the parabola intersect. Since we found one unique solution at (-1, -3), this means that the line and the parabola intersect at exactly one point, which is (-1, -3). This graphical interpretation reinforces our algebraic solution and provides a visual understanding of the problem.

Alternative Methods for Solving Systems of Equations

While we used the substitution method to solve this particular system of equations, it's important to be aware of other methods that can be used. These include:

  • Elimination Method: This method involves manipulating the equations to eliminate one variable, allowing you to solve for the other. It is particularly useful when the coefficients of one variable are multiples of each other.
  • Graphing Method: As discussed earlier, this method involves plotting the equations on a graph and finding the points of intersection. It is a visual method that can be helpful for understanding the nature of the solutions.
  • Matrix Methods: For systems with more than two variables, matrix methods such as Gaussian elimination or matrix inversion can be used to solve the system.

Choosing the most appropriate method depends on the specific system of equations. For simple systems like the one we solved, the substitution method is often the most straightforward approach. However, for more complex systems, other methods may be more efficient.

Key Takeaways and Conclusion

In this article, we addressed the problem of finding the solutions to a system of equations. We walked through the process of solving the system:

-4x - 7 = y
x² - 2x - 6 = y

using the substitution method. We found that there is one unique real number solution at (x, y) = (-1, -3). This solution was confirmed by analyzing the algebraic steps and considering the graphical interpretation of the equations.

Key Takeaways:

  • To solve a system of equations, find the values of the variables that satisfy all equations simultaneously.
  • The substitution method is a useful technique for solving systems where one variable is expressed in terms of the other.
  • Factoring, completing the square, and the quadratic formula are methods for solving quadratic equations.
  • The number of real number solutions corresponds to the number of intersection points of the equations' graphs.
  • There is one unique real number solution for David's system of equation.

By understanding the methods and concepts discussed in this article, you will be well-equipped to solve a variety of systems of equations and analyze the nature of their solutions. Remember to practice and explore different approaches to enhance your problem-solving skills.

Solving Systems of Equations Find Real Number Solutions Step by Step

Find the solutions to the system of equations.

Solving Systems of Equations A Comprehensive Guide to Finding Real Number Solutions