Solving System Of Linear Equations An In-Depth Analysis
In the realm of mathematics, particularly in linear algebra, the concept of systems of linear equations holds significant importance. These systems, comprised of two or more equations involving the same set of variables, arise in various fields, from engineering and physics to economics and computer science. Understanding how to solve these systems and interpret their solutions is crucial for problem-solving and modeling real-world phenomena. This comprehensive exploration delves into a specific system of linear equations, meticulously examining its properties and unveiling the nature of its solution. We will dissect the equations, analyze their graphical representation, and determine the characteristics of their intersection. By the end of this analysis, we will have a clear understanding of the system's behavior and the implications of its solution.
The system of linear equations we will be focusing on is as follows:
Our objective is to determine which of the following statements accurately describe the system:
- A. The system has one solution.
- B. The system graphs parallel lines.
- C. Both equations represent the same line.
To embark on this exploration, we will first manipulate the equations into a more standard form, making it easier to analyze their properties. This involves isolating the 'y' variable on one side of each equation, thereby transforming them into the slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. This transformation will allow us to readily compare the slopes and y-intercepts of the two lines, providing valuable insights into their relationship and the nature of their intersection.
Transforming the Equations into Slope-Intercept Form
The first step in our analysis is to convert the given equations into the slope-intercept form, which is represented as y = mx + b, where m denotes the slope and b represents the y-intercept. This form provides a clear representation of the line's inclination and its point of intersection with the y-axis. By transforming the equations into this form, we can readily compare their slopes and y-intercepts, gaining valuable insights into their relationship and the nature of their solution.
Let's begin by manipulating the first equation:
2y = x + 10
To isolate y, we divide both sides of the equation by 2:
y = (1/2)x + 5
Now, let's transform the second equation:
3y = 3x + 15
Similarly, we divide both sides of the equation by 3 to isolate y:
y = x + 5
By transforming the equations into slope-intercept form, we have successfully expressed them in a format that allows for easy comparison. The first equation, y = (1/2)x + 5, has a slope of 1/2 and a y-intercept of 5. The second equation, y = x + 5, has a slope of 1 and a y-intercept of 5. These values provide crucial information about the lines' orientations and their point of intersection with the y-axis. The differences in their slopes indicate that the lines are not parallel, while the shared y-intercept suggests a common point on the y-axis. This initial analysis lays the groundwork for a more comprehensive understanding of the system's solution.
Analyzing the Slopes and Y-Intercepts
Having successfully transformed the equations into slope-intercept form, we now possess the necessary tools to analyze their slopes and y-intercepts. This analysis is crucial in determining the relationship between the two lines and the nature of their solution. The slope, denoted by m in the equation y = mx + b, dictates the line's steepness and direction. A steeper slope indicates a more rapid change in y for a given change in x, while the sign of the slope determines whether the line rises or falls as x increases. The y-intercept, denoted by b, represents the point where the line intersects the y-axis. This point provides a fixed reference on the coordinate plane, anchoring the line's position.
Comparing the slopes of the two equations, we observe that the first equation, y = (1/2)x + 5, has a slope of 1/2, while the second equation, y = x + 5, has a slope of 1. These slopes are distinct, indicating that the lines are not parallel. Parallel lines, by definition, have the same slope and will never intersect. Since the slopes in our system are different, we can confidently conclude that the lines will intersect at some point. This intersection point represents the solution to the system of equations, the values of x and y that satisfy both equations simultaneously.
Furthermore, we notice that both equations share the same y-intercept of 5. This means that both lines intersect the y-axis at the same point, (0, 5). While this shared y-intercept provides a common point on both lines, it does not imply that the lines are identical. The differing slopes ensure that the lines will diverge from this point, eventually intersecting at another location. The combination of distinct slopes and a shared y-intercept paints a clear picture of two lines that intersect at a single, unique point, solidifying our understanding of the system's solution.
Determining the Number of Solutions
The number of solutions a system of linear equations possesses is directly related to the graphical representation of the equations. When two lines intersect at a single point, the system has one unique solution, corresponding to the coordinates of that intersection point. If the lines are parallel, they never intersect, and the system has no solution. Conversely, if the two equations represent the same line, they overlap completely, and the system has infinitely many solutions, as every point on the line satisfies both equations.
In our analysis of the slopes and y-intercepts, we established that the two lines in our system have different slopes but the same y-intercept. This configuration indicates that the lines are neither parallel nor identical. The differing slopes guarantee that the lines will intersect at a single point, while the shared y-intercept provides a common point on both lines. This unique intersection point represents the solution to the system, the specific values of x and y that satisfy both equations simultaneously.
To find this solution, we can use several methods, such as substitution or elimination. However, for the purpose of determining the number of solutions, the graphical analysis is sufficient. The fact that the lines intersect at one point confirms that the system has one unique solution. This solution represents the coordinates of the intersection point, the values of x and y that make both equations true. Therefore, we can confidently conclude that statement A, "The system has one solution," is a true statement about the system of linear equations.
Evaluating the Statements
Having meticulously analyzed the system of linear equations, we are now equipped to evaluate the given statements and determine their veracity. Each statement presents a specific characteristic of the system, and our analysis has provided the necessary insights to assess their accuracy. By carefully considering the slopes, y-intercepts, and graphical representation of the equations, we can confidently identify the statements that accurately describe the system's properties.
Statement A: "The system has one solution." As we have established through our analysis of the slopes and y-intercepts, the two lines in the system intersect at a single point. This intersection point represents the unique solution to the system, the values of x and y that satisfy both equations. Therefore, statement A is indeed a true statement.
Statement B: "The system graphs parallel lines." Parallel lines, by definition, have the same slope and never intersect. In our analysis, we found that the two lines have different slopes, indicating that they are not parallel. The differing slopes guarantee that the lines will intersect at some point, making statement B a false statement.
Statement C: "Both equations represent the same line." For two equations to represent the same line, they must have the same slope and the same y-intercept. While the two equations in our system share the same y-intercept, they have different slopes. This difference in slopes implies that the lines will diverge from their common y-intercept, eventually intersecting at another point. Therefore, the equations do not represent the same line, and statement C is a false statement.
In conclusion, our comprehensive analysis reveals that only statement A accurately describes the system of linear equations. The system has one solution, corresponding to the intersection point of the two lines. Statements B and C are false, as the lines are neither parallel nor identical. This thorough evaluation demonstrates the power of mathematical analysis in understanding the properties and solutions of linear systems.
Conclusion: One Unique Solution
In summary, our comprehensive exploration of the system of linear equations
has led us to a definitive conclusion. By transforming the equations into slope-intercept form, we were able to meticulously analyze their slopes and y-intercepts. This analysis revealed that the lines have distinct slopes but share the same y-intercept, indicating that they intersect at a single, unique point. This intersection point represents the sole solution to the system, the specific values of x and y that satisfy both equations simultaneously.
Our evaluation of the given statements confirmed that only statement A, "The system has one solution," is true. Statements B and C, which claimed that the lines are parallel or identical, were found to be false based on our analysis of the slopes and y-intercepts. The differing slopes definitively ruled out the possibility of parallel lines, while the shared y-intercept, coupled with the differing slopes, negated the possibility of the equations representing the same line.
This detailed analysis underscores the importance of understanding the fundamental properties of linear equations and their graphical representations. By mastering the concepts of slope, y-intercept, and the relationship between lines, we can effectively solve systems of equations and interpret their solutions. The ability to analyze and solve linear systems is a valuable skill in various fields, enabling us to model and understand real-world phenomena that can be represented by linear relationships. The unique solution we found in this system highlights the power of mathematical tools in providing precise and accurate answers to complex problems.
