Analyzing The Solution Of A System Of Linear Equations
In this article, we will delve into the system of equations presented and analyze its characteristics to determine the correct statement about its solution. The system of equations we will be examining is:
y = 2x - 3
y = -3
We will explore the slopes, intercepts, and graphical representation of these equations to arrive at a conclusive answer. Understanding the nature of solutions to systems of linear equations is a fundamental concept in algebra, with applications spanning various fields, including engineering, economics, and computer science. Therefore, a thorough examination of this system is crucial for grasping these underlying principles.
Understanding Linear Equations
To begin our analysis, it's essential to understand the fundamental components of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The standard form of a linear equation is often expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope (m) indicates the steepness and direction of the line, while the y-intercept (b) is the point where the line crosses the y-axis.
In our system of equations, the first equation, y = 2x - 3, is in slope-intercept form. This form allows us to quickly identify the slope and y-intercept. By comparing it to the standard form, we can see that the slope (m) is 2 and the y-intercept (b) is -3. This means that for every one unit increase in x, the value of y increases by two units. The line crosses the y-axis at the point (0, -3).
The second equation, y = -3, is a special case of a linear equation. It represents a horizontal line. In this case, the slope is 0, as the value of y remains constant regardless of the value of x. The y-intercept is -3, meaning the line passes through the point (0, -3).
Understanding the slopes and y-intercepts of these equations is crucial for determining the relationship between the lines they represent and, ultimately, the solution to the system of equations.
Analyzing the Slopes and Intercepts
Now that we have established the basic understanding of linear equations, let's dive deeper into the specific characteristics of our given system:
y = 2x - 3
y = -3
As previously identified, the first equation, y = 2x - 3, has a slope of 2 and a y-intercept of -3. The second equation, y = -3, has a slope of 0 and a y-intercept of -3. Comparing the slopes, we can clearly see that they are different. The first line has a non-zero slope, indicating it is slanted, while the second line has a slope of 0, making it a horizontal line. The difference in slopes is a significant factor in determining the nature of the solution to the system.
However, observing the y-intercepts, we notice that both lines share the same y-intercept of -3. This means that both lines intersect the y-axis at the same point (0, -3). The fact that the lines have different slopes but share a y-intercept provides crucial information about their graphical representation and the solution to the system.
To further visualize this, consider the graphical representation of these lines. A line with a slope of 2 will rise steeply as x increases, while the horizontal line y = -3 remains constant. The point where these two lines intersect represents the solution to the system of equations. In this case, since they have different slopes and share a y-intercept, we can anticipate a single, unique solution.
Graphical Representation and Solutions
Visualizing the equations graphically can provide a clearer understanding of their relationship and the solution to the system. The equation y = 2x - 3 represents a line that slopes upwards from left to right. To plot this line, we can start at the y-intercept (0, -3) and use the slope of 2 to find another point. Moving one unit to the right and two units up, we reach the point (1, -1). Connecting these two points gives us the graph of the first equation.
The equation y = -3 represents a horizontal line that passes through all points where the y-coordinate is -3. This line intersects the y-axis at (0, -3) and extends horizontally across the coordinate plane.
When we graph both lines on the same coordinate plane, we can clearly see their point of intersection. The point of intersection represents the solution to the system of equations, as it is the only point that satisfies both equations simultaneously. In this case, the two lines intersect at the point (0, -3). This intersection visually confirms that the system has a single, unique solution.
If the lines were parallel, they would never intersect, indicating that there is no solution to the system. If the lines were the same, they would overlap completely, indicating an infinite number of solutions. However, in our case, the distinct slopes and the single point of intersection confirm that there is exactly one solution.
Determining the Correct Statement
Having analyzed the system of equations in detail, we can now evaluate the given statements to determine which one is true. The system of equations is:
y = 2x - 3
y = -3
Let's consider the possible statements:
A. The lines have different slopes. B. There is no solution to the system. C. The lines have the same y-intercept.
From our analysis, we know that the slopes of the lines are 2 and 0, respectively. Therefore, statement A, “The lines have different slopes,” is true. The lines are not parallel, and they will intersect at one specific point.
Statement B, “There is no solution to the system,” is incorrect. As we discussed and visualized graphically, the lines intersect at a single point, indicating that there is a unique solution. The fact that the lines have different slopes ensures that they will intersect.
Statement C, “The lines have the same y-intercept,” while true (both lines have a y-intercept of -3), is not the primary factor determining the nature of the solution in this case. The crucial aspect is the difference in slopes, which guarantees a unique solution. While the shared y-intercept contributes to the point of intersection, the differing slopes are the key to the system's solvability.
Therefore, based on our comprehensive analysis, the correct statement about the system of linear equations is that the lines have different slopes. This difference in slopes leads to the existence of a single, unique solution to the system.
Conclusion
In conclusion, by thoroughly examining the system of equations y = 2x - 3 and y = -3, we have determined that the statement “The lines have different slopes” is the true statement. We achieved this conclusion by analyzing the slopes and intercepts of the equations, visualizing their graphical representation, and understanding the relationship between these properties and the solution to the system.
This exercise highlights the importance of understanding the fundamental concepts of linear equations, including slope, y-intercept, and graphical representation. These concepts are crucial for solving systems of equations and for applying mathematical principles to real-world problems. The ability to analyze and interpret linear equations is a valuable skill in various fields, from mathematics and science to economics and engineering.
By recognizing the different slopes and the implications for the lines' intersection, we were able to correctly identify the true statement about the system. This systematic approach to problem-solving, involving analysis, visualization, and logical deduction, is a powerful tool for tackling mathematical challenges.
Therefore, the final answer is A. The lines have different slopes.
