Understanding Slope-Intercept Form: Finding the Equation of a Line with Given Slope and Intercept

Slope-intercept form is a fundamental concept in algebra and graphing, allowing us to easily identify the slope and y-intercept of a line. This form is particularly useful for quick graphing and determining the equation of a line when given specific parameters. Let's dive into a detailed explanation of how to find the equation of a line in slope-intercept form when the slope and y-intercept are provided.

What is Slope-Intercept Form?

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

y mx b

In this formula:

m: the slope of the line b: the y-intercept, which is the point where the line crosses the y-axis (i.e., the value of y when x 0)

Given Slope and Y-Intercept

Suppose we are given the slope and the y-intercept of a line. The equation of the line in slope-intercept form can be directly written as:

y mx b

For example, if the slope (m) is 5 and the y-intercept (b) is -2, the equation becomes:

y 5x - 2

So, the equation of the line in slope-intercept form is:

y 5x - 2

Understanding the Given Question

The question asks for an equation in slope-intercept form for the line with slope 5 and y-intercept -2. Given this information:

Slope (m) 5 Y-intercept (b) -2

We can directly write the equation as:

y 5x - 2

However, if the "r-intercept" in the question is intended to be the "y-intercept," then the equation is:

y 5x - 2

Special Cases: Finding Y-Intercept from X-Intercept

What if the problem modifies the question slightly, asking for the line with slope 5 and x-intercept -2? In this case, the x-intercept is the point where the line crosses the x-axis (i.e., the value of x when y 0). Let's solve this step by step:

Slope (m) 5 X-intercept: -20 (this suggests that the line passes through (-20, 0))

Using the slope-intercept form, we start with:

0 5(-20) b

Solving for (b):

0 -100 b

b 100

Therefore, the equation of the line is:

y 5x 100

While this example slightly deviates from the original problem statement, it highlights the importance of understanding different forms of linear equations and how to manipulate them based on given conditions.

Conclusion

Understanding slope-intercept form is crucial for anyone studying algebra and graphing linear equations. Whether you are given both the slope and the y-intercept directly or asked to find the y-intercept from the x-intercept, being able to write the equation in slope-intercept form is a valuable skill. Practice with these types of problems will help solidify your understanding and improve your graphing abilities.