Equation of the Straight Line: Finding Slope and Y-Intercept

Determining the equation of a straight line that passes through specific points is a fundamental concept in mathematics. This article will guide you through the process of finding the slope and y-intercept of a line passing through the points (2, 3) and (-4, 9).

Background

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

y mx c

Where m is the slope of the line, and c is the y-intercept. To determine these values, we start with the given points and apply the formulas for slope and y-intercept.

Step 1: Calculate the Slope (m)

The slope m of a line passing through two points ((x_1, y_1)) and ((x_2, y_2)) is calculated using the formula:

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

Let's substitute the points (2, 3) and (-4, 9) into the formula:

x_1 2, y_1 3, x_2 -4, y_2 9

m frac{9 - 3}{-4 - 2} frac{6}{-6} -1

Step 2: Find the Y-Intercept (c)

Now that we have the slope, we can use one of the given points to find the y-intercept. Let's use the point (2, 3):

3 -1(2) c

3 -2 c

c 3 2 5

Conclusion

The values of m and c are:

m -1

c 5

Therefore, the equation of the line is:

y -x 5

This can also be written in standard form as:

x y - 5 0

Common Methods and Points to Check

Using the point (-4, 9) to check the equation:

9 -(-4) 5

9 4 5

9 9

Both points (2, 3) and (-4, 9) satisfy the equation, confirming that it is correct.

Conclusion

By finding the slope and y-intercept, we can determine the equation of a line passing through two given points. This method can be applied to any pair of coordinates, allowing us to solve a wide range of linear equation problems.