Finding the Slope-Intercept Form of a Line Given its X- and Y-Intercepts

When dealing with equations of lines in geometry and algebra, it's crucial to understand how to express a line in different forms, including the slope-intercept form. This form provides a clear visualization of the line's slope (m) and its y-intercept (b): y mx b. In this article, we will explore how to find the slope-intercept form of a line given its x-intercept and y-intercept. Specifically, we will solve for a line that intersects the x-axis at 5 and the y-axis at 3.

Introduction to X- and Y-Intercepts

Before delving into the equation, it's important to understand what x- and y-intercepts are. The x-intercept is the point where a line crosses the x-axis, meaning the y-coordinate at that point is 0. Thus, it can be represented as (a, 0). Similarly, the y-intercept is the point where a line crosses the y-axis, meaning the x-coordinate is 0. This point can be represented as (0, b).

Given Information and Equation Setup

We are given that the x-intercept is 5 and the y-intercept is 3. This means the line can be represented as:

[ frac{x}{5} frac{y}{3} - 1 0 ]

To convert this into the slope-intercept form, we need to manipulate the equation to isolate y on one side.

Converting to Slope-Intercept Form

The standard form equation we have is:

[ frac{x}{5} frac{y}{3} - 1 0 ]

To convert this into slope-intercept form, let's first add 1 to both sides:

[ frac{x}{5} frac{y}{3} 1 ]

Next, we can eliminate the fractions by multiplying every term by 15 (the least common multiple of 5 and 3):

[ 3x 5y 15 ]

Now, to isolate y, subtract 3x from both sides:

[ 5y -3x 15 ]

Finally, divide every term by 5 to solve for y:

[ y -frac{3}{5}x 3 ]

This is the equation of the line in slope-intercept form. Here, (-frac{3}{5} ) is the slope (m) and 3 is the y-intercept (b).

Explanation of the Slope-Intercept Form

The slope-intercept form ( y mx b ) is particularly useful because it provides an easy way to graph the line. The slope (m) tells us the rate at which y changes for each unit change in x, and the y-intercept (b) gives us the point where the line crosses the y-axis.

In our given example, the slope is (-frac{3}{5} ), meaning for every increase of 5 units in x, y decreases by 3 units. The y-intercept is 3, so the line crosses the y-axis at the point (0, 3).

Conclusion and Application

Understanding how to convert a line's equation from standard form to slope-intercept form is a fundamental skill in algebra and geometry. It helps in various applications, such as linear programming, financial mathematics, and engineering.

By knowing the x-intercept and y-intercept, we can not only determine the equation of the line but also graph it easily. This method is not only useful for math problems but also for real-world applications where linear relationships are involved, such as determining the cost of a product based on quantity sold or analyzing trends in data.

Remember, practice is the key to mastering these concepts. Try solving similar problems with different intercept values to reinforce your understanding.

Keywords: slope-intercept form, x-intercept, y-intercept