Determining the Equation of the Straight Line Passing Through Points P and Q

In this article, we will walk through the process of determining the equation of a straight line passing through two given points, P(5, 1) and Q(4, 5). This involves understanding the concept of slope and the point-slope form of a line equation. By the end of this guide, you will have a clear understanding of how to find the equation of a line and how to apply it in real-world scenarios.

Understanding the Concept of Slope

The slope of a straight line, denoted by the letter 'm', is a measure of the line's steepness. It is calculated as the change in the y-coordinates divided by the change in the x-coordinates between any two points on the line. Mathematically, the slope can be expressed as:

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

For the points P(5, 1) and Q(4, 5), the slope can be calculated as follows:

[ m frac{5 - 1}{4 - 5} frac{4}{-1} -4 ]

The negative slope indicates that the line is decreasing from left to right.

Using the Point-Slope Form to Find the Equation

The point-slope form of a line equation is particularly useful when you know the slope and one point on the line. The general form is:

[ y - y_1 m(x - x_1) ]

Here, (m) is the slope, and ((x_1, y_1)) is a point on the line. Using the slope (m -4) and point P(5, 1), we can substitute these values into the point-slope form:

[ y - 1 -4(x - 5) ]

Now, we will simplify this equation to find its standard form.

Deriving the Standard Form of the Line Equation

Starting with the point-slope form:

[ y - 1 -4(x - 5) ]

We can distribute the -4:

[ y - 1 -4x 20 ]

Next, we will isolate y by adding 1 to both sides:

[ y -4x 21 ]

Therefore, the equation of the straight line passing through points P and Q in standard form is:

[ y -4x 21 ]

Conclusion

In this article, we have gone through the step-by-step process of determining the equation of a straight line using two given points. The key concept is understanding the slope and applying the point-slope form of the line equation. This method is not only useful for mathematical problems but also in various real-world applications such as physics, engineering, and data analysis.

If you need further assistance or have more questions, feel free to explore additional resources or seek professional help from experienced mathematicians or educators.

Remember, practice is key to mastering these concepts. Try solving similar problems with different sets of points to reinforce your understanding.

Keywords: straight line equation, slope, point-slope form