Equation of a Straight Line Passing Through Given Points

In this article, we will explore how to determine the equation of a straight line that passes through two given points. We will use a specific example, where the line passes through the points (-2, 4) and (5, -3), to illustrate the process step-by-step. We will start by finding the slope of the line and then proceed to derive the equation in point-slope form and convert it to a more familiar form.

Understanding the Slope (Rate of Change)

The slope of a line, denoted by m, is a measure of the rate of change of y with respect to x. It is calculated using the formula:

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

For the points (-2, 4) and (5, -3), we can calculate the slope as follows:

m frac{-3 - 4}{5 - (-2)}

m frac{-7}{7}

m -1

This value of m represents the constant rate of change of the line, which is -1 in this case.

Point-Slope Form of the Equation of a Line

The point-slope form of the equation of a line is given by:

y - y_1 m(x - x_1)

where (x_1, y_1) is a point on the line and m is the slope. Let's use the point (-2, 4) to write the equation:

y - 4 -1(x - (-2))

y - 4 -1(x 2)

y - 4 -x - 2

y -x - 2 4

y -x 2

Thus, the equation of the line in slope-intercept form is y -x 2.

Conclusion

Understanding the slope and using the point-slope form are crucial steps in determining the equation of a straight line passing through two given points. In this case, the line passing through (-2, 4) and (5, -3) has the equation y -x 2.

By following these steps, you can determine the equation of any straight line given two points on the line. This knowledge is vital for various applications in mathematics and real-world scenarios, including graphing and solving geometric problems.

Keywords

slope point-slope form equation of a line

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