Solving The System Of Equations 2y-x=6 And X=-2y+6 A Comprehensive Guide

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Introduction: Understanding Systems of Equations

In the realm of mathematics, particularly in algebra, solving systems of equations is a fundamental skill. 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, when substituted for the variables, make all the equations true. These systems arise in various real-world applications, from physics and engineering to economics and computer science. Understanding how to solve them is crucial for problem-solving in these fields. This article delves into a specific system of linear equations, providing a step-by-step guide to finding its solution and discussing the implications of the result.

The system we will be focusing on is:

2y - x = 6
x = -2y + 6

This system consists of two linear equations with two variables, x and y. Linear equations, characterized by variables raised to the power of one, represent straight lines when graphed on a coordinate plane. The solution to a system of linear equations corresponds to the point(s) where the lines intersect. There are several methods to solve such systems, including substitution, elimination, and graphing. We will primarily use the substitution method in this article, as it is particularly well-suited for this system.

Before diving into the solution, it’s important to understand the different types of solutions a system of linear equations can have:

  1. Unique Solution: The lines intersect at exactly one point, indicating a single, unique solution for x and y. This is the most common scenario.
  2. No Solution: The lines are parallel and never intersect, meaning there is no solution that satisfies both equations simultaneously. This occurs when the lines have the same slope but different y-intercepts.
  3. Infinitely Many Solutions: The lines are coincident, meaning they are essentially the same line. Every point on the line represents a solution, leading to an infinite number of solutions. This happens when the equations are multiples of each other.

Determining which of these scenarios applies to a given system is a key part of the problem-solving process. In the following sections, we will systematically solve the given system and interpret the results.

Step-by-Step Solution Using Substitution

The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be easily solved. 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.

Let's apply the substitution method to our system:

2y - x = 6  (Equation 1)
x = -2y + 6 (Equation 2)

Notice that Equation 2 is already solved for x. This makes the substitution method particularly straightforward in this case. We can directly substitute the expression for x from Equation 2 into Equation 1:

2y - (-2y + 6) = 6

Now, we have a single equation with only one variable, y. Let's simplify and solve for y:

2y + 2y - 6 = 6
4y - 6 = 6
4y = 12
y = 3

We have found that y = 3. Now, we can substitute this value back into either Equation 1 or Equation 2 to find the value of x. Equation 2 is simpler, so let's use that:

x = -2(3) + 6
x = -6 + 6
x = 0

Therefore, we have found that x = 0. Our solution to the system is the ordered pair (0, 3). This means that the point (0, 3) is the intersection point of the two lines represented by the equations in the system.

To verify our solution, we can substitute both x = 0 and y = 3 back into both original equations:

For Equation 1:

2(3) - 0 = 6
6 = 6 (True)

For Equation 2:

0 = -2(3) + 6
0 = -6 + 6
0 = 0 (True)

Since the solution (0, 3) satisfies both equations, we have confirmed that it is the correct solution to the system. In the next section, we will discuss the nature of this solution and what it implies about the relationship between the two lines.

Analyzing the Solution and System Type

Having found the solution (0, 3) for the system of equations, it's crucial to analyze what this solution tells us about the nature of the system. As mentioned earlier, a system of linear equations can have a unique solution, no solution, or infinitely many solutions. Our result of (0, 3) indicates that this system has a unique solution. This means that the two lines represented by the equations intersect at exactly one point on the coordinate plane, which is the point (0, 3).

To further understand the system, let's rewrite both equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. This form provides valuable insights into the characteristics of the lines.

Starting with Equation 1:

2y - x = 6
2y = x + 6
y = (1/2)x + 3

The slope of the first line is 1/2, and the y-intercept is 3.

Now, let's rewrite Equation 2:

x = -2y + 6
2y = -x + 6
y = (-1/2)x + 3

There appears to be a mistake in the original equation. It should be 2y = -x + 6, leading to y = (-1/2)x + 3. However, if we proceed with the provided equation x = -2y + 6, let's rearrange it to the slope-intercept form. To do this, we'll solve for y:

x = -2y + 6 2y = -x + 6 y = -1/2 x + 3

This is crucial because, without this transformation, comparing the slopes and y-intercepts directly is impossible. From the slope-intercept form, several key aspects of the line become immediately apparent, such as its steepness and where it crosses the y-axis.

Let's correct the analysis with the correct transformation:

The slope of the second line is -1/2, and the y-intercept is 3.

Now, we can make some observations:

  1. Different Slopes: The two lines have slopes of 1/2 and -1/2, respectively. Since the slopes are different, the lines are not parallel and will intersect at some point. This confirms that there is a unique solution.
  2. Same y-intercept: Both lines have a y-intercept of 3. This means that both lines cross the y-axis at the same point, (0, 3). This is precisely the solution we found earlier, which makes sense because the point of intersection must lie on both lines.

The fact that the lines have different slopes guarantees a unique solution. If the slopes were the same, the lines would either be parallel (no solution) or coincident (infinitely many solutions). In this case, the distinct slopes and the shared y-intercept uniquely define the intersection point, confirming our solution of (0, 3).

This analysis provides a deeper understanding of the system of equations and its solution. It's not enough to simply find the solution; understanding the geometric interpretation of the equations and the implications of the solution is equally important. In the next section, we will explore alternative methods for solving this system and discuss the advantages and disadvantages of each.

Alternative Methods for Solving the System

While the substitution method proved to be quite effective for solving the system of equations,

2y - x = 6
x = -2y + 6

it's beneficial to explore alternative methods to broaden our problem-solving toolkit. Two common methods are elimination and graphing.

1. Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations in the system so that the coefficients of one variable are opposites. When the equations are added together, that variable is eliminated, leaving a single equation with one variable. This method is particularly useful when the equations are in standard form (Ax + By = C).

To apply the elimination method to our system, we can rearrange Equation 2 to match the form of Equation 1:

x = -2y + 6
x + 2y = 6

Now we have the system:

2y - x = 6
x + 2y = 6

Notice that the coefficients of x are already opposites (-1 and 1). We can add the two equations directly:

(2y - x) + (x + 2y) = 6 + 6
4y = 12
y = 3

We obtain y = 3, which is the same value we found using the substitution method. Now, we can substitute y = 3 into either of the original equations to find x. Let's use Equation 2:

x = -2(3) + 6
x = -6 + 6
x = 0

Again, we find x = 0. Thus, the solution using the elimination method is also (0, 3), confirming our previous result.

The elimination method is often preferred when the equations are in standard form or when it's easy to create opposite coefficients by multiplying one or both equations by a constant.

2. Graphing Method

The graphing method involves plotting the lines represented by the equations on a coordinate plane. The solution to the system is the point(s) where the lines intersect. This method provides a visual representation of the system and its solution.

We already rewrote the equations in slope-intercept form earlier:

y = (1/2)x + 3
y = (-1/2)x + 3

To graph these lines, we can use the slope and y-intercept. Both lines have a y-intercept of 3, so they both pass through the point (0, 3). The first line has a slope of 1/2, meaning for every 2 units we move to the right, we move 1 unit up. The second line has a slope of -1/2, meaning for every 2 units we move to the right, we move 1 unit down.

Plotting these lines on a graph, we would see that they intersect at the point (0, 3). This visually confirms our solution.

The graphing method is useful for understanding the geometric interpretation of the system and for estimating the solution. However, it may not be the most accurate method for finding exact solutions, especially if the solution involves non-integer values.

Each of these methods – substitution, elimination, and graphing – has its strengths and weaknesses. The choice of method often depends on the specific system of equations and personal preference. In the case of our system, both substitution and elimination were relatively straightforward due to the structure of the equations. The graphing method provided a visual confirmation of the solution.

Conclusion: The Significance of Solving Systems of Equations

In this article, we've undertaken a comprehensive exploration of a specific system of linear equations:

2y - x = 6
x = -2y + 6

We successfully solved this system using the substitution method, arriving at the unique solution (0, 3). This solution represents the point of intersection of the two lines defined by the equations. We then verified our solution by substituting the values back into the original equations.

Furthermore, we analyzed the nature of the system by rewriting the equations in slope-intercept form. This analysis revealed that the lines have different slopes (1/2 and -1/2) and the same y-intercept (3). The differing slopes confirmed that the system has a unique solution, while the shared y-intercept pinpointed the y-coordinate of the intersection point.

To broaden our understanding, we explored alternative methods for solving the system, including the elimination method and the graphing method. The elimination method provided an algebraic approach that corroborated our solution, while the graphing method offered a visual confirmation of the intersection point. Each method has its advantages and is suitable for different types of systems.

The ability to solve systems of equations is a cornerstone of mathematical proficiency. It extends far beyond the classroom and into numerous real-world applications. Systems of equations are used to model and solve problems in diverse fields such as:

  • Physics: Analyzing motion, forces, and circuits.
  • Engineering: Designing structures, optimizing processes, and controlling systems.
  • Economics: Modeling supply and demand, predicting market trends, and optimizing resource allocation.
  • Computer Science: Developing algorithms, solving optimization problems, and creating simulations.
  • Everyday Life: Making informed decisions about budgeting, travel planning, and resource management.

Mastering the techniques for solving systems of equations empowers individuals to tackle complex problems and make data-driven decisions. The concepts learned in this context, such as variable manipulation, equation solving, and graphical interpretation, are transferable skills that enhance problem-solving abilities in various domains.

In conclusion, the exploration of the system {2y - x = 6, x = -2y + 6} has provided a valuable case study for understanding systems of linear equations. Through the application of multiple solution methods and a thorough analysis of the results, we have gained a deeper appreciation for the significance of this fundamental mathematical concept. Whether you are a student, a professional, or simply someone seeking to enhance their problem-solving skills, mastering systems of equations is an investment that yields significant returns.