Determining the Slope of a Line Given a Point and Y-Intercept

In this article, we will explore how to determine the slope of a line when given a specific point and its y-intercept. We will go through several methods to find the slope of a line, ensuring that the instructions are clear and easy to follow.

Introduction to Slopes and Y-Intercepts

Before we delve into the methods, let's briefly review what a slope and y-intercept are. The slope of a line is a measure of its steepness and is calculated as the change in y divided by the change in x. The y-intercept is the point where the line crosses the y-axis.

Method 1: Using the Slope-Intercept Form

The slope-intercept form of a line is given by:

y mx b

where m is the slope and b is the y-intercept. We are given that the y-intercept b is 5. Therefore, the equation of the line can be written as:

y mx - 5

We also know that the line passes through the point (2, 3). By substituting the coordinates (2, 3) into the equation, we can solve for the slope m:

Substitute x 2 and y 3 into the equation: 3 m(2) - 5 Rearranging the equation gives: 3 5 2m Subtract 5 from both sides: 8 2m Divide both sides by 2: m -1

Therefore, the slope of the line is -1.

Method 2: Using the Two-Point Formula

We can also find the slope of the line using the two-point formula:

m frac{y_2 - y_1}{x_2 - x_1}

We are given the point (2, 3) and the y-intercept (0, 5). By substituting these points into the formula, we can find the slope:

Substitute (2, 3) as (x_1, y_1) and (0, 5) as (x_2, y_2) into the formula: m frac{5 - 3}{0 - 2} simplify the expression: m frac{2}{-2} Calculate the slope: m -1

Therefore, the slope of the line is -1.

Conclusion

In conclusion, there are multiple methods to find the slope of a line given a point and y-intercept. Whether you choose to use the slope-intercept form or the two-point formula, both methods will lead you to the same result. Understanding these methods is essential for solving problems related to linear equations and geometry.

References

[1] MathIsFun. Slope.