Equation of a Straight Line through Given Points: A Comprehensive Guide

In this guide, you will learn how to determine the equation of a straight line that passes through two given points. We will walk through the process step-by-step, providing examples and detailing the key concepts involved.

Understanding the Problem

Given two points, (x1, y1) and (x2, y2), such as (41, -35), you can use the slope formula and the point-slope form of the line equation to find the equation of the straight line.

Step 1: Calculate the Slope (m)

The formula to calculate the slope (m) is given by:

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

Let's apply this formula using the points (41, -35) and (4, 1):

[ m frac{1 - (-35)}{4 - 41} frac{36}{-37} -frac{36}{37} ]

However, due to the given example, let's correct it to match the provided points:

[ m frac{-35 - 1}{4 - 41} frac{-36}{-37} frac{36}{37} ]

Note: The provided example does not match the points (41, -35) and (4, 1), so the correct slope is (-frac{4}{7}).

Step 2: Use the Point-Slope Form

The point-slope form of the line equation is:

[ y - y_1 m(x - x_1) ]

Using the point (4, 1) and the corrected slope (-frac{4}{7}), we get:

[ y - 1 -frac{4}{7}(x - 4) ]

Let's simplify this equation:

[ y - 1 -frac{4}{7}x frac{16}{7} ]

[ y -frac{4}{7}x frac{16}{7} 1 ]

( y -frac{4}{7}x frac{16}{7} frac{7}{7} )

( y -frac{4}{7}x frac{23}{7} )

Step 3: Simplify the Equation

Convert the equation to the standard form (Ax By C 0):

[ y -frac{4}{7}x frac{23}{7} ]

Multiply through by 7 to eliminate the fractions:

[ 7y -4x 23 ]

[ 4x 7y - 23 0 ]

Conclusion

The equation of the straight line passing through the points (41, -35) and (4, 1) is: