Solving Systems Of Equations A Step-by-Step Guide

Are you struggling with solving systems of equations? This comprehensive guide will walk you through the process, providing a step-by-step solution to the given system and offering valuable insights into the underlying concepts. Our primary focus is to help you understand how to solve such problems efficiently and accurately. Let's dive into the world of algebra and master this essential skill.

Understanding Systems of Equations

Before we tackle the specific problem, it's crucial to understand what a system of equations is. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Graphically, the solution represents the point(s) where the graphs of the equations intersect. For linear equations, this intersection is a single point, but for non-linear equations like the quadratics we’ll be dealing with, there may be multiple points of intersection, or none at all.

In this case, we are given a system of two quadratic equations. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is two. These equations often represent parabolas when graphed, and their solutions can be found using various algebraic methods, such as substitution, elimination, or factoring. Solving a system of quadratic equations requires finding the points where these parabolas intersect, which can lead to real-world applications in fields like physics and engineering, where parabolic trajectories are common.

When approaching a system of equations, it’s important to identify the type of equations involved, as this will guide the choice of solution method. For systems involving linear and quadratic equations, substitution is often the most straightforward approach. For systems involving only linear equations, methods like elimination or matrix operations may be more efficient. Understanding the properties of different types of equations is essential for choosing the best strategy to solve systems of equations effectively. By mastering these techniques, you can tackle a wide range of mathematical problems and gain a deeper understanding of algebraic relationships.

The Given System of Equations

The system of equations we need to solve is:

y = x^2 + 10x + 11
y = x^2 + x - 7

This system consists of two quadratic equations, both expressed in terms of y. Our goal is to find the values of x and y that satisfy both equations simultaneously. To achieve this, we will use the substitution method, which is a powerful technique for solving systems of equations. Substitution involves solving one equation for one variable and substituting that expression into the other equation. This process eliminates one variable, allowing us to solve for the remaining variable. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

The presence of x² in both equations suggests that these are quadratic equations, meaning they will likely have parabolic graphs. When two parabolas intersect, they can do so at zero, one, or two points. This means that our system may have no solution, one solution, or two solutions. By solving the system algebraically, we will determine the exact number of solutions and their values. The intersection points represent the pairs of (x, y) values that satisfy both equations, providing a complete solution to the system. Understanding the graphical representation of these equations helps visualize the solution process and provides a deeper understanding of the algebraic manipulations we perform. Solving systems of equations is not just about finding numbers; it’s about understanding the relationships between equations and their geometric interpretations.

Step-by-Step Solution

1. Setting the Equations Equal

Since both equations are expressed in terms of y, we can set them equal to each other:

x^2 + 10x + 11 = x^2 + x - 7

This step is crucial because it eliminates y from the equation, allowing us to focus solely on x. By equating the expressions for y, we are essentially finding the x-values where the two parabolas have the same y-value, which corresponds to the points of intersection. This method is a direct application of the substitution principle, where we replace one variable with an equivalent expression to simplify the equation. The resulting equation is a single-variable equation that can be solved using standard algebraic techniques.

The strategic advantage of this approach is that it transforms a system of two equations into a single equation, making it more manageable. This is a common technique used in solving systems of equations, particularly when one or more equations are already solved for a variable. By carefully choosing which variable to eliminate, we can often simplify the problem and reduce the amount of algebraic manipulation required. Understanding the underlying principles of substitution allows for a more intuitive and efficient solution process. Solving systems of equations often involves identifying the most effective method to reduce complexity, and equating expressions is a powerful tool in that regard.

2. Simplifying the Equation

Now, let's simplify the equation by subtracting x² from both sides:

10x + 11 = x - 7

This simplification is a key step in solving systems of equations as it transforms the quadratic equation into a linear equation. The x² terms cancel each other out, making the equation much easier to solve. This outcome often occurs when dealing with systems where the quadratic terms are identical in both equations. By eliminating the quadratic term, we reduce the complexity of the problem and move closer to isolating the variable x. Linear equations are straightforward to solve using basic algebraic operations, making this simplification a significant milestone in the solution process.

By subtracting x² from both sides, we maintain the equality of the equation while making it more manageable. This is a fundamental principle in algebra – performing the same operation on both sides of an equation preserves the balance and ensures that the solution remains valid. The resulting linear equation allows us to use simple techniques such as adding, subtracting, multiplying, and dividing to isolate the variable. Recognizing these simplification opportunities is crucial for efficient solving systems of equations, as it streamlines the process and minimizes the chances of errors. Mastering these simplification techniques is essential for tackling more complex algebraic problems in the future.

3. Isolating x

Next, subtract x from both sides:

9x + 11 = -7

This step is a crucial part of isolating the variable x, which is a fundamental technique in solving systems of equations. By subtracting x from both sides, we move all terms involving x to one side of the equation, while keeping the constant terms on the other side. This separation allows us to further simplify the equation and eventually solve for x. The principle behind this step is to maintain the balance of the equation while strategically rearranging terms to achieve our goal.

Subtracting x from both sides ensures that the equation remains equivalent to the original. This is a basic algebraic manipulation that is essential for solving various types of equations, not just systems of equations. The goal is to isolate x by undoing the operations that are being performed on it. In this case, we are undoing the addition of x on the right side of the equation. By continuing this process of isolating x, we will eventually arrive at a solution that satisfies the given system of equations. Solving systems of equations often involves a series of such manipulations, each designed to bring us closer to the final answer.

4. Continuing to Isolate x

Now, subtract 11 from both sides:

9x = -18

Continuing the process of isolating x, this step is crucial in solving systems of equations. By subtracting 11 from both sides, we further simplify the equation by isolating the term with x on one side and the constant term on the other. This operation maintains the balance of the equation while moving us closer to our goal of finding the value of x. The strategic rearrangement of terms is a common technique used in algebraic manipulations to solve for unknowns.

Subtracting 11 from both sides is a straightforward arithmetic operation, but its impact on the equation is significant. It removes the constant term from the left side, allowing us to focus solely on the term containing x. This is a standard step in solving linear equations and is part of a broader strategy for solving systems of equations. By systematically isolating x, we are setting the stage for the final step, which will reveal the value of x. Understanding and applying these basic algebraic operations is essential for mastering equation-solving techniques.

5. Solving for x

Finally, divide both sides by 9:

x = -2

This step concludes the isolation of x, a critical component in solving systems of equations. By dividing both sides of the equation by 9, we determine the value of x that satisfies the simplified linear equation. This solution for x is a key piece of information, as it represents the x-coordinate of the point where the two parabolas intersect. The division operation is the inverse of multiplication, and it effectively undoes the multiplication of x by 9, leaving x by itself on one side of the equation.

The value x = -2 is a significant result because it provides one part of the solution to the system of equations. To find the complete solution, we still need to determine the corresponding value of y. However, knowing x is a major step forward in solving systems of equations, as it allows us to substitute this value back into one of the original equations to find y. The process of isolating a variable and solving for its value is a fundamental skill in algebra, and it is essential for solving a wide range of mathematical problems.

6. Substituting x to Find y

Now that we have x = -2, we can substitute this value into either of the original equations to find y. Let's use the second equation:

y = (-2)^2 + (-2) - 7

This substitution is a crucial step in solving systems of equations, as it allows us to determine the corresponding value of y for the x-value we found. By substituting x = -2 into the second equation, we are essentially finding the y-coordinate of the point where the two parabolas intersect. This process ensures that we find a pair of (x, y) values that satisfy both equations simultaneously, which is the definition of a solution to a system of equations.

The choice of which equation to substitute into is often a matter of convenience, as both equations should yield the same value for y. However, selecting the simpler equation can sometimes reduce the amount of computation required. In this case, the second equation appears slightly simpler, making it a practical choice. The act of substituting and solving for the remaining variable is a fundamental technique in solving systems of equations, and it highlights the interconnectedness of the variables within the system.

7. Solving for y

Simplify the equation:

y = 4 - 2 - 7
y = -5

This step completes the process of finding the y-value, which is essential in solving systems of equations. By simplifying the equation, we calculate the value of y that corresponds to the x-value we previously found. This calculation involves basic arithmetic operations and follows directly from the substitution step. The result, y = -5, represents the y-coordinate of the intersection point between the two parabolas defined by the original equations.

The simplification process is straightforward but crucial for accuracy. Each arithmetic operation must be performed carefully to avoid errors that could lead to an incorrect solution. The final value of y is a critical component of the solution, as it provides the complete coordinates of the point that satisfies both equations in the system. With both x and y values determined, we have successfully found a solution to the system of equations. The ability to accurately solve for y after finding x is a key skill in solving systems of equations and demonstrates a thorough understanding of algebraic principles.

The Solution

The solution to the system of equations is the ordered pair (x, y) = (-2, -5).

This ordered pair represents the point where the two parabolas intersect, and it is the only pair of values that satisfies both equations simultaneously. The solution is a concise and complete answer to the problem, providing both the x and y coordinates of the intersection point. Verifying the solution by substituting these values back into the original equations is a good practice to ensure accuracy in solving systems of equations.

The ordered pair notation is a standard way to represent solutions to systems of equations, clearly indicating the relationship between the x and y values. The solution (-2, -5) means that when x is -2, y is -5 in both equations. This unique pair of values is the key to solving systems of equations and represents a fundamental concept in algebra. The solution is not just a set of numbers; it’s a specific point in the coordinate plane that satisfies the conditions set by the equations.

Identifying the Correct Option

Looking at the given options, the correct answer is:

A. (-2, -5)

This confirms that our step-by-step solution has led us to the correct answer. Identifying the correct option is the final step in the problem-solving process, and it demonstrates a clear understanding of how to solve systems of equations and interpret the results. The ability to match the solution to the provided options is an important skill, especially in standardized tests and assessments.

The option (-2, -5) aligns perfectly with our calculated solution, reinforcing the accuracy of our algebraic manipulations and problem-solving approach. This final verification step ensures that we have not only found a solution but also correctly identified it among the given choices. Solving systems of equations involves not only the algebraic process but also the ability to interpret and present the solution in the appropriate format.

Conclusion

In this comprehensive guide, we have thoroughly explained the process of solving systems of equations, specifically focusing on a system of two quadratic equations. We walked through each step, from setting the equations equal to each other to simplifying and isolating variables, and finally, identifying the correct solution.

The key takeaways from this guide include the importance of understanding the underlying principles of algebraic manipulation, the strategic use of the substitution method, and the ability to accurately perform arithmetic operations. Solving systems of equations is a fundamental skill in mathematics, and mastering this process opens the door to more advanced topics and real-world applications. By practicing these techniques and understanding the concepts, you can confidently tackle similar problems and enhance your problem-solving abilities.

Remember, the solution to a system of equations represents the point(s) of intersection between the graphs of the equations. In the case of quadratic equations, this often involves parabolas, and the solutions correspond to the points where the parabolas intersect. Understanding this graphical interpretation provides a deeper understanding of the algebraic processes involved in solving systems of equations. Keep practicing, and you'll become proficient in solving a wide range of algebraic problems.