How to Find the Equation of a Straight Line Through Given Points

Determining the equation of a straight line that passes through specific points is a fundamental skill in algebra and can be crucial in various applications. In this guide, we will walk through a step-by-step process to find the equation of a straight line that passes through the points (3, 9) and (-6, -36).

Methodology

To find the equation of a straight line that passes through two points, we can use the following approach:

Step 1: Calculate the Slope

The slope m of a line passing through two points (x?, y?) and (x?, y?) is given by the formula:
m frac{y_2 - y_1}{x_2 - x_1}
Using the points (3, 9) and (-6, -36):
x? 3, y? 9
x? -6, y? -36
Substituting these values into the slope formula:

m  frac{-36 - 9}{-6 - 3}  frac{-45}{-9}  5

Step 2: Use the Point-Slope Form of the Equation

With the slope m 5 and one of the points (3, 9), we can use the point-slope form of the equation of a line:
y - y_1 m(x - x_1)
Substituting in the values:

y - 9  5(x - 3)

Step 3: Simplify to Slope-Intercept Form

Distributing the slope on the right side:

y - 9  5x - 15

Now add 9 to both sides:

y  5x - 6

The final equation of the straight line that passes through the points (3, 9) and (-6, -36) is:

y  5x - 6

Verification

To verify the equation, we can plug in the coordinates of the points to ensure they satisfy the equation:

When x 3, y 5(3) - 6 9 When x -6, y 5(-6) - 6 -36

This confirms that the equation y 5x - 6 is correct.

Alternative Methods

Here are some alternative methods to find the equation of a straight line that passes through the points (3, 9) and (-6, -36):

Using the general form: Ax By C. In this case:

Substitute the points into the equation to find A, B, and C:

9  3A   B-36  -6A - B

Add the two equations to eliminate B:

45  9AA  5

Substitute A back into one of the equations to solve for B:

9  3(5)   BB  -6

The equation in general form is:

5x - y - 6  0

Summary

In summary, finding the equation of a straight line that passes through given points involves calculating the slope, using the point-slope form, and converting to the slope-intercept form. This method is widely applicable and can be a powerful tool in various mathematical and real-world applications.

Key takeaways:

Slope of a line: m frac{y_2 - y_1}{x_2 - x_1} Point-slope form: y - y_1 m(x - x_1) Slope-intercept form: y mx b