Introduction to Finding the Slope of a Line Through Given Points

In this comprehensive guide, we will delve into the process of finding the slope of a line that passes through two given points. Specifically, we will explore the method using the slope formula and provide step-by-step solutions for finding the slope of the line that passes through the points (4, 2) and (6, 7).

Slope Formula and Its Application

The slope (m) of a line that passes through two points (x1, y1) and (x2, y2) is calculated using the formula:

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

For the points (4, 2) and (6, 7), the slope calculation is as follows:

Step 1: Identify the Coordinates

Let:

x1 4, y1 2 x2 6, y2 7

Step 2: Substitute into the Formula

Substituting the coordinates into the slope formula:

m (frac{7 - 2}{6 - 4})

m (frac{5}{2})

Hence, the slope of the line passing through the points (4, 2) and (6, 7) is (frac{5}{2}).

Alternative Method: Using Linear Equations

Alternatively, we can determine the equation of the line passing through these points by using the slope-intercept form:

y mx c

Where:

m is the slope we just calculated, (frac{5}{2}) c is the y-intercept, which we need to determine

Determining the y-intercept (c)

Substitute the coordinates of one of the points into the equation to solve for c.

Using the point (4, 2):

2 (frac{5}{2}) * 4 c

Solving for c:

2 10 c

c 2 - 10 -8

Therefore, the equation of the line is:

y (frac{5}{2})x - 8

Additional Methods and Interpretations

Let us interpret the solution using another approach. By subtracting the equations formed by the two points, we can find the slope directly:

Step 1: Form Equations

Equation 1: 2 4m c (using point (4, 2))

Equation 2: 7 6m c (using point (6, 7))

Step 2: Subtract Equations

Equation 2 - Equation 1:

7 - 2 6m - 4m

5 2m

m (frac{5}{2})

This confirms that the slope is (frac{5}{2}).

Conclusion

In conclusion, the slope of the line passing through the points (4, 2) and (6, 7) is (frac{5}{2}). Whether using the slope formula directly, solving linear equations, or employing subtraction techniques, the result is consistent and robust.

Understanding these methods not only strengthens your grasp of linear algebra but also enhances your problem-solving skills in various applications, such as graphing, data analysis, and more.