Graphing a Line with Slope -1 Through Point (4, 5): A Comprehensive Guide

Understanding how to graph a line with a specific slope passing through a given point is a fundamental skill in algebra and graphing. In this article, we will walk you through the process step-by-step to graph a line with a slope of -1 that passes through the point (4, 5). By the end, you will have a clear understanding of the concepts involved and be able to apply them to similar problems.

Step-by-Step Guide to Graphing the Line

Step 1: Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation:

y mx b

where m is the slope and b is the y-intercept. This form is useful because it directly gives us the slope and the y-intercept of the line, which are crucial for graphing.

Step 2: Identify the Slope and Point

In this case, the slope m is -1, and the point we are given is (4, 5). Therefore, we have:

m -1

Point (4, 5)

Step 3: Use the Point to Find the Y-Intercept

To find the y-intercept, we substitute the point (4, 5) and the slope -1 into the slope-intercept form equation:

5 -1(4) b

Solving for b gives us:

5 -4 b

b 5 4

b 9

Step 4: Write the Equation of the Line

Now that we have the y-intercept, we can write the equation of the line:

y -1x 9 or y -x 9

Step 5: Graph the Line

Plot the Point (4, 5)

Starting with the given point (4, 5), plot this point on the graph.

Find Another Point Using the Slope

Since the slope is -1, we can move down 1 unit and to the right 1 unit from the point (4, 5) to find another point:

To the right 1 unit: 5 1 6

Down 1 unit: 5 - 1 4

This gives us the point (5, 4).

Draw the Line

Connect the points (4, 5) and (5, 4) with a straight line. Extend the line in both directions, ensuring it continues infinitely, and add arrows at both ends to indicate continuity.

Step 6: Check the Y-Intercept

As a verification, we can check the y-intercept by setting x 0 in the equation:

y -0 9

y 9

This confirms that the line intersects the y-axis at (0, 9).

Summary

The line with a slope of -1 passing through the point (4, 5) can be represented by the equation y -x 9 or y -x 9. To graph this line, first plot the given point (4, 5) and use the slope to find additional points. Then, draw a straight line through these points, extending it in both directions.

Additional Tips

Understanding the slope-intercept form of a line and how to use it to graph lines is essential. Remember that the slope represents both the direction (positive or negative) and the steepness of the line. The y-intercept is the point where the line crosses the y-axis.

Practice is key to mastering these skills. Try graphing other lines with different slopes and points to reinforce your understanding.