Understanding the Slope Intercept Form of a Line Through Point (2, -1) with a Slope of -3

When dealing with linear equations in a coordinate plane, we often encounter the need to determine the slope-intercept form of the line. The slope-intercept form of a line is a fundamental concept in algebra and is expressed as:

y mx b

Where:

m is the slope of the line b is the y-intercept of the line, which is the value of y when x 0

Slope-Intercept Form and Given Conditions

Given the slope of a line as -3 and a point (2, -1) that lies on the line, we can use the slope-intercept formula to find the equation of the line. The general approach involves substituting the known values and solving for the y-intercept.

The slope m is -3. The point (2, -1) must satisfy the equation y -3x b.

Deriving the Y-Intercept

To find the y-intercept b, we can use the given point (2, -1) and the slope -3. Substituting these values into the equation:

-1 -3(2) b

Let's perform the calculations step by step:

-1 -6 b -1 6 b b 5

Therefore, the slope-intercept form of the line is:

y -3x 5

Summary and Graphical Explanation

Let's summarize the key steps:

Slope m -3 Point (2, -1) Solving for y-intercept b 5

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

y -3x 5

Geometric Interpretation

For a point (2, -1) and a slope of -3, the y-intercept is 5. This means that the line will intersect the y-axis at the point (0, 5).

To understand this geometrically, consider the point 2, -1 moving vertically from (2, -1) to (0, 5), which is a vertical distance of 6 units upward. Since the slope is -3, this indicates a decrease in the y-values by 3 for every unit increase in the x-values.

Conclusion

A line with a slope of -3 passing through the point (2, -1) can be expressed in the slope-intercept form as:

y -3x 5

Understanding this form is crucial for various applications in mathematics, such as graphing, solving equations, and analyzing linear relationships in real-world scenarios.