Graphing Linear Systems Which Graph Represents 2x - 5y = -5 And Y = -2/5x + 1

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In the realm of mathematics, particularly within the domain of linear algebra, understanding how to visually represent systems of equations is a fundamental skill. This article delves into the process of identifying the graph that corresponds to a given system of linear equations. We will specifically focus on the system defined by the equations 2x - 5y = -5 and y = -2/5x + 1. By carefully analyzing the equations, determining their slopes and y-intercepts, and considering their potential intersection, we can accurately pinpoint the graph that embodies this system.

Understanding the Equations

Before we dive into the graphical representation, it's crucial to dissect the given equations and extract key information. The first equation, 2x - 5y = -5, is in standard form. To better understand its behavior, we can transform it into slope-intercept form (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. Let's perform this conversion:

  1. Subtract 2x from both sides: -5y = -2x - 5
  2. Divide both sides by -5: y = (2/5)x + 1

Now, we have the first equation in slope-intercept form. We can readily identify its slope as 2/5 and its y-intercept as 1. This means the line rises 2 units for every 5 units it runs to the right, and it crosses the y-axis at the point (0, 1).

The second equation, y = -2/5x + 1, is already in slope-intercept form. Its slope is -2/5, and its y-intercept is also 1. This line falls 2 units for every 5 units it runs to the right, and it intersects the y-axis at the same point (0, 1) as the first line.

Analyzing Slopes and Intercepts

The slopes and y-intercepts we've determined are critical clues in identifying the correct graph. Notice that the two lines have different slopes (2/5 and -2/5). This signifies that they are not parallel; they will intersect at some point. Furthermore, both lines share the same y-intercept (1), meaning they intersect the y-axis at the exact same location.

The fact that the slopes have opposite signs also tells us something important: one line will have a positive slope (rising from left to right), while the other will have a negative slope (falling from left to right). This visual characteristic should be evident in the correct graph.

Identifying the Graph

With a clear understanding of the slopes, y-intercepts, and the relationship between the lines, we can now systematically analyze potential graphs. Here's a breakdown of the key features to look for:

  • Two distinct lines: The graph must depict two separate lines, representing the two equations in the system.
  • Y-intercept at (0, 1): Both lines should intersect the y-axis at the point where y equals 1.
  • Opposite slopes: One line should have a positive slope (rising from left to right), and the other should have a negative slope (falling from left to right).
  • Intersection point: The lines should intersect at some point, indicating a solution to the system of equations. Since they share the same y-intercept, this point of intersection will be (0, 1).

By visually inspecting graphs and comparing them to these criteria, you can confidently identify the one that accurately represents the system of equations 2x - 5y = -5 and y = -2/5x + 1.

The Significance of Graphical Representation

The graphical representation of a system of equations provides a powerful visual tool for understanding the relationship between the equations and their solutions. The point of intersection, if it exists, represents the solution to the system – the set of x and y values that satisfy both equations simultaneously. In this specific case, since the lines intersect at (0, 1), this point is the unique solution to the system.

Furthermore, the graphical representation can reveal important information about the nature of the system. For instance:

  • Intersecting lines: Indicate a unique solution, as seen in this example.
  • Parallel lines: Indicate no solution, as the lines never intersect.
  • Coincident lines: Indicate infinitely many solutions, as the lines overlap completely.

Understanding these graphical interpretations is crucial for solving systems of equations and gaining a deeper understanding of linear relationships.

To further solidify your understanding, let's walk through the process of graphing the system of equations 2x - 5y = -5 and y = -2/5x + 1 from scratch. This step-by-step guide will not only help you visualize the equations but also reinforce the concepts of slope, y-intercept, and intersection.

1. Prepare the Equations

As we discussed earlier, it's beneficial to have the equations in slope-intercept form (y = mx + b) for easy graphing. We've already converted the first equation:

  • 2x - 5y = -5 becomes y = (2/5)x + 1

The second equation is already in slope-intercept form:

  • y = -2/5x + 1

Now, we have both equations in the desired format, ready for plotting.

2. Identify Slopes and Y-intercepts

For each equation, extract the slope (m) and the y-intercept (b):

  • Equation 1: y = (2/5)x + 1
    • Slope (m) = 2/5
    • Y-intercept (b) = 1
  • Equation 2: y = -2/5x + 1
    • Slope (m) = -2/5
    • Y-intercept (b) = 1

These values are the foundation for plotting the lines.

3. Plot the Y-intercepts

The y-intercept is the point where the line crosses the y-axis. For both equations, the y-intercept is 1. This means both lines will pass through the point (0, 1) on the coordinate plane. Mark this point for both lines.

4. Use the Slope to Find Additional Points

The slope (m) represents the