Understanding the Slope-Intercept Form for a Line with Slope 1/3 and Y-Intercept -5

When working with linear equations, the slope-intercept form is a fundamental concept. This form is particularly useful because it allows us to easily identify the slope and the y-intercept of a line. For a line with a slope of 1/3 and a y-intercept of -5, we can express the equation in the slope-intercept form as follows:

The Formula and Explanation

The general formula for the slope-intercept form is:

y mx b

Where m represents the slope of the line. b represents the y-intercept, which is the point where the line crosses the y-axis.

Applying the Given Values

In our case, the slope (m) is given as 1/3, and the y-intercept (b) is -5. Plugging these values into the formula, we get:

y (1/3)x - 5

Verification and Simplification

Another way to express the same equation is to write it as:

y (1/3)x - 5

This can also be expressed using the point-slope form:

y 5 (1/3)(x - 0)

Which simplifies to:

y (1/3)x - 5

Graphical Representation

When graphing this line, we start by plotting the y-intercept at the point (0, -5). The slope of 1/3 indicates that for every 3 units you move to the right, you move up 1 unit. This helps in plotting additional points on the line for a clearer representation.

Common Applications

The slope-intercept form is widely used in various fields such as physics, economics, and engineering to model linear relationships. For example, in physics, it can be used to describe the relationship between distance and time for an object moving at a constant velocity. In economics, it might represent the relationship between price and quantity demanded.

Conclusion

Understanding the slope-intercept form for a line is crucial for various mathematical and practical applications. For a line with a slope of 1/3 and a y-intercept of -5, the equation of the line in slope-intercept form is:

y (1/3)x - 5