Understanding the Equation of a Line with a Specific Slope and Y-Intercept

When given the slope and y-intercept of a line, we can easily write its equation using the slope-intercept form. This form is particularly useful for visualizing and solving various problems related to linear equations.

Introduction to Slope and Y-Intercept

In the context of linear equations, the slope (denoted as ( m )) represents the rate of change of the line. It tells us how steep the line is. The y-intercept (denoted as ( c )) is the point where the line crosses the y-axis. In other words, it's the value of ( y ) when ( x 0 ).

Slope-Intercept Form of a Line

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

( y mx c )

Example Problem

Let’s consider a specific example to illustrate the application of the slope-intercept form.

Given:

Slope (m) -4 y-intercept (c) 3

Using the slope-intercept form, we can write:

( y -4x 3 )

This equation tells us that for every unit increase in ( x ), ( y ) decreases by 4 units. The line crosses the y-axis at the point (0, 3).

Alternative Forms of Line Equations

While the slope-intercept form is straightforward, there are other forms of line equations that can be useful in different scenarios. For instance, the point-slope form:

( y - y_1 m(x - x_1) )

Derivation from Given Slope and Y-Intercept

Using the given slope ( m -4 ) and y-intercept ( c -3 ), let’s derive the equation of the line using the point-slope form:

Identify the y-intercept as the point (0, -3). Apply the point-slope form equation:

( y - (-3) -4(x - 0) )

Which simplifies to:

( y 3 -4x )

And further simplifies to:

( y -4x - 3 )

Visualization and Graphical Interpretation

To better understand the relationship between slope and y-intercept, let's consider a graphical representation:

The y-intercept is the point where the line crosses the y-axis. In this case, it's (0, -3). The slope indicates the rate of change, and -4 means that for every one unit increase in ( x ), ( y ) decreases by 4 units.

Another key point to consider is the x-intercept. To find the x-intercept, set ( y 0 ) and solve for ( x ):

( 0 -4x - 3 )

Solving this equation, we get:

( x -frac{3}{4} )

So the x-intercept is ( left( -frac{3}{4}, 0 right) ).

Conclusion

Understanding the equation of a line with a specific slope and y-intercept is crucial for solving various mathematical and real-world problems. By using the slope-intercept form, we can efficiently write and interpret the equation of a line. The point-slope form provides an alternative method for deriving the equation, further enhancing our understanding of linear relationships.