Graphing the Line y -2x - 5: A Comprehensive Guide

When working with linear equations in algebra, one of the most fundamental tasks is to graph the line represented by the equation. The line y -2x - 5 is such an equation, and it can be plotted by determining a few key points and connecting them. This guide will walk you through the process step-by-step, using simple procedures and clear examples.

Understanding the Equation

The equation y -2x - 5 is in the slope-intercept form, y mx b, where m is the slope, and b is the y-intercept. For this equation:
- m -2, which means the slope is -2. This indicates that for every 1 unit increase in x, y decreases by 2 units.
- b -5, which is the y-intercept. This is the point where the line crosses the y-axis, which in this case is at point (0, -5).

Plotting the Y-Intercept

The first step in graphing the line y -2x - 5 is to plot the y-intercept, (0, -5). This means that the line will pass through the y-axis at the point where x 0 and y -5. To plot this point, simply mark the point where the y-axis intersects with the line at -5 on the y-axis.

Using the Slope to Find Another Point

The next step is to use the slope to find another point on the line. The slope, -2, tells us that for every 1 unit increase in x, y decreases by 2 units. Starting from the y-intercept, (0, -5): Move 1 unit to the right (to x 1). Move 2 units down (to y -7). So, the next point on the line is (1, -7).

Connecting the Points with a Ruler

With two points, (0, -5) and (1, -7), you can now connect them with a ruler. Extend the line in both directions, and add arrows to indicate that it continues infinitely. This will give you the complete graph of the line y -2x - 5.

Alternative Methods

While using the y-intercept and slope is a common method, you can also calculate other points to ensure accuracy. For example: When x -3: Calculate y -2(-3) - 5 6 - 5 1. The point is (-3, 1). When x 5: Calculate y -2(5) - 5 -10 - 5 -15. The point is (5, -15). Plot these points and connect them to verify the line's accuracy.

Alternatively, you can also start at the y-intercept (0, -5) and use the slope to find points. Since the slope is -2, moving to the right by 1 unit and down by 2 units will give you the next point, and so on.

Conclusion

In summary, to graph the line y -2x - 5, follow these steps: Plot the y-intercept, (0, -5). Use the slope to find another point. Draw a straight line through these points, extending in both directions. Graphing lines in algebra is a fundamental skill, and understanding these methods will help you tackle more complex equations and concepts in mathematics.