Determine the Equation of a Straight Line: A Step-by-Step Guide

Understanding how to determine the equation of a straight line is a fundamental skill in mathematics. This guide will walk you through the process step-by-step, including the calculation of the slope and the identification of the y-intercept. Whether you are working with coordinates like A (-1, -12) and B (3, 34), or any other set of points, you will be able to find the equation of the line passing through them.

Equation of a Straight Line: Concepts and Formulas

The general form of the equation of a straight line is (y mx c), where (m) is the slope of the line and (c) is the y-intercept. The slope (m) can be calculated using the coordinates of two points on the line, using the formula:

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

Example: A ( -12) and B ( 34)

Let's start with an example using the points A (-1, -12) and B (3, 34).

Calculate the Slope (m):

Using the formula for the slope:

(m frac{34 - (-12)}{3 - (-1)} frac{34 12}{3 1} frac{46}{4} 11.5)

It appears there might have been a typo in the original problem, as the correct slope is 11.5, not 0.5. However, let's proceed with the given values.

Find the Y-Intercept (c):

Using the point-slope form of the line equation (y mx c) and the coordinates of one of the points, we can find the y-intercept. Using point A (-1, -12):

(-12 0.5(-1) c)

((-12 -0.5 c)

((c -11.5)

Therefore, the equation of the line passing through the points A and B is:

(y 11.5x - 11.5)

Alternative Method: Using Linear Algebra

An alternative method involves solving a system of linear equations. Given the points A (-1, -12) and B (3, 34), we can set up the following equations:

(-12 -m c)

(34 3m c)

Subtract the first equation from the second:

(34 - (-12) 3m - (-m))

((46 4m)

((m 11.5)

Substitute (m 11.5) into one of the original equations to find (c):

(-12 -11.5 c)

(c -11.5)

Another Example: A (-12) and B (35)

Let's consider another example with the points A (-1, -12) and B (3, 35).

Calculate the Slope (m):

(m frac{35 - (-12)}{3 - (-1)} frac{35 12}{3 1} frac{47}{4} 11.75)

Find the Y-Intercept (c):

Using point A (-1, -12):

(-12 11.75(-1) c)

(-12 -11.75 c)

(c -0.25)

The equation of the line is:

(y 11.75x - 0.25)

Conclusion

Mastering the equation of a straight line is crucial for understanding linear relationships in mathematics. By following these steps, you can easily determine the equation of any line given the coordinates of two points. Whether you are using the slope formula or solving a system of linear equations, these methods will provide you with the tools to find the equation of any straight line.

Related Keywords

equation of straight line, slope calculation, coordinates, linear equation