Solving Systems Of Equations A Step By Step Guide

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In mathematics, 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 that satisfy all equations simultaneously. Finding these solutions is a fundamental concept in algebra, with applications spanning various fields like engineering, economics, and computer science. In this comprehensive guide, we will explore a system of equations and walk through the process of determining the solution. We will use the substitution method to solve the given system of equations, providing a clear and concise explanation for each step.

Understanding the System of Equations

Our journey begins with a clear understanding of the equations we are dealing with. The system presented is:

y = 4x - 10
y = 2

This system consists of two linear equations, each representing a straight line on a coordinate plane. The solution we seek is the point where these two lines intersect, if such a point exists. This point of intersection represents the (x, y) values that satisfy both equations. To fully grasp the nature of solutions to systems of equations, it's crucial to recognize the three possibilities that can arise:

  1. Unique Solution: The lines intersect at a single point, indicating a unique solution where the x and y values satisfy both equations simultaneously.
  2. No Solution: The lines are parallel and never intersect, signifying that there are no values of x and y that can satisfy both equations. This implies the system is inconsistent.
  3. Infinite Solutions: The lines are coincident, meaning they overlap completely. In this case, every point on the line represents a solution, as the equations are essentially the same, just expressed differently.

Before diving into the solution process, let's reinforce why solving systems of equations is so vital. It's not just an abstract mathematical exercise; it's a tool that unlocks solutions to real-world problems. Imagine modeling the supply and demand curves in economics – the equilibrium point, where supply equals demand, is found by solving a system of equations. Or consider designing a bridge, where the forces and stresses must balance; systems of equations are used to ensure structural integrity. In computer graphics, transformations like rotations and scaling are expressed as matrix equations, and solving these systems is crucial for rendering images correctly. Understanding the nature of solutions – unique, none, or infinite – is equally important, as it tells us whether a problem has a single answer, no answer, or a range of possibilities.

Solving the System Using Substitution

In this particular scenario, the substitution method offers an efficient way to find the solution. The substitution method is a powerful technique used to solve systems of equations by expressing one variable in terms of another and substituting that expression into another equation. This process reduces the system to a single equation with a single variable, which can then be solved directly.

Step 1: Identify an equation where one variable is already isolated.

Looking at the system, the second equation y = 2 readily provides the value of y. This simplifies our task significantly, as we know that the y-coordinate of the solution will be 2.

Step 2: Substitute the known value into the other equation.

Now, we substitute the value of y (which is 2) into the first equation:

2 = 4x - 10

This substitution effectively eliminates y from the first equation, leaving us with an equation in terms of x only. This is the core idea behind the substitution method: to reduce the system to a single equation with a single unknown.

Step 3: Solve for the remaining variable (x).

We now have a simple linear equation to solve for x. To isolate x, we first add 10 to both sides of the equation:

2 + 10 = 4x - 10 + 10
12 = 4x

Next, we divide both sides by 4:

12 / 4 = 4x / 4
3 = x

Therefore, we find that x = 3.

Step 4: Express the solution as an ordered pair.

We have now found the values of both x and y that satisfy the system of equations. We found that x = 3 and y = 2. The solution is typically expressed as an ordered pair (x, y), which in this case is (3, 2).

Each step in the substitution method is crucial, and understanding the underlying logic ensures that the solution is obtained accurately. Identifying the isolated variable (or isolating one if necessary) is the first key step, as it sets the stage for the substitution. The substitution itself is the heart of the method, as it reduces the complexity of the system. Solving for the remaining variable is a straightforward algebraic process, and expressing the solution as an ordered pair provides a clear and concise representation of the solution.

Verifying the Solution

To ensure accuracy, it is always recommended to verify the solution by substituting the values of x and y back into the original equations. This step acts as a check, confirming that the values indeed satisfy both equations simultaneously.

Substitute x = 3 and y = 2 into the first equation:

y = 4x - 10
2 = 4(3) - 10
2 = 12 - 10
2 = 2

The first equation holds true.

Substitute x = 3 and y = 2 into the second equation:

y = 2
2 = 2

The second equation also holds true.

Since the solution (3, 2) satisfies both equations, we can confidently confirm that it is the correct solution to the system of equations. This verification step is not merely a formality; it's a crucial practice that safeguards against errors and reinforces understanding of the solution process. By plugging the values back into the original equations, we are essentially reversing the steps of the solution, ensuring that the logical flow is consistent and that no algebraic mistakes were made. In more complex systems, verification becomes even more critical, as the chances of error increase. It's a habit that every student of algebra should cultivate.

Identifying the Correct Answer

Now, let's examine the given answer choices:

a. (3, 2) b. (2, 3) c. (-2, 2) d. (2, -2)

Comparing our solution (3, 2) with the answer choices, we can clearly see that option a. (3, 2) is the correct answer.

This final step of identifying the correct answer underscores the importance of careful and accurate execution throughout the solution process. Each step, from understanding the system of equations to verifying the solution, contributes to the overall accuracy. A mistake at any stage can lead to selecting the wrong answer. Therefore, attention to detail, a systematic approach, and a thorough understanding of the underlying principles are paramount.

Alternative Methods for Solving Systems of Equations

While we have successfully solved this system using the substitution method, it is important to acknowledge that there are other methods available for solving systems of equations. Each method has its own strengths and weaknesses, and the most appropriate method often depends on the specific system of equations being solved.

  1. Elimination Method: The elimination method involves manipulating the equations in the system (by multiplying by constants) such that the coefficients of one of the variables are opposites. When the equations are added together, that variable is eliminated, leaving a single equation in one variable. This method is particularly useful when the equations are in standard form (Ax + By = C).

  2. Graphing: Graphing involves plotting the equations on a coordinate plane. The solution to the system is the point(s) where the graphs intersect. This method provides a visual representation of the solution and is useful for understanding the nature of the solutions (unique, none, or infinite). However, graphing may not be as accurate for finding exact solutions, especially if the solutions are not integers.

  3. Matrix Methods: Matrix methods, such as Gaussian elimination and using the inverse of a matrix, are powerful techniques for solving larger systems of equations. These methods are particularly well-suited for computer implementation and are used extensively in various fields.

Understanding these alternative methods not only expands our problem-solving toolkit but also provides a deeper appreciation for the interconnectedness of mathematical concepts. Each method offers a different perspective on the same problem, and choosing the most efficient method can save time and effort. For instance, in our example, the substitution method was particularly efficient due to the presence of an already isolated variable. However, for a system where neither variable is isolated, the elimination method might be more straightforward.

Conclusion

In this comprehensive guide, we have successfully solved the given system of equations using the substitution method. We meticulously walked through each step, from understanding the system to verifying the solution, and identified the correct answer choice. We also briefly explored other methods for solving systems of equations, highlighting the versatility of mathematical tools. Mastering the solution of systems of equations is a crucial step in mathematical development, with wide-ranging applications in various fields. By understanding the underlying concepts and practicing diligently, you can confidently tackle a wide range of problems involving systems of equations.

Keywords: system of equations, substitution method, solution, linear equations, elimination method, graphing, matrix methods, ordered pair, verify solution, algebra.