Introduction to the Problem

This article provides a step-by-step guide on how to find the equation of a line that is perpendicular to another line and passes through a given point. This process involves understanding the concepts of slope, the slope of a perpendicular line, and the point-slope form of a line equation.

Understanding the Problem

The problem requires us to find the equation of the line that passes through the point (-1, 3) and is perpendicular to the line passing through points (-3, 5) and (2, 1).

Step 1: Finding the Slope of the Given Line

To find the slope of the line passing through the points (-3, 5) and (2, 1), we use the slope formula:

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

Substituting the given points (-3, 5) and (2, 1), we get:

m frac{1 - 5}{2 - (-3)} frac{-4}{5}

The slope m of the given line is ( -frac{4}{5} ).

Step 2: Finding the Slope of the Perpendicular Line

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, if the original slope is ( -frac{4}{5} ), the slope of the perpendicular line ( m_{perp} ) is:

m_{perp} -frac{1}{m} -frac{1}{-frac{4}{5}} frac{5}{4}

The slope of the perpendicular line is ( frac{5}{4} ).

Step 3: Using the Point-Slope Form of the Line Equation

We use the point-slope form of the equation of a line, which is given by:

y - y_1 m(x - x_1)

Here, we use the point (-1, 3) and the slope ( frac{5}{4} ) to substitute into the equation:

y - 3 frac{5}{4}(x - (-1))

y - 3 frac{5}{4}x frac{5}{4}

Step 4: Rearranging to Slope-Intercept Form

Now, we rearrange the equation to the slope-intercept form, which is given by:

y mx b

Substituting the terms, we get:

y - 3 frac{5}{4}x frac{5}{4}

y frac{5}{4}x frac{5}{4} 3

y frac{5}{4}x frac{5}{4} frac{12}{4}

y frac{5}{4}x frac{17}{4}

Thus, the equation of the line passing through the point (-1, 3) and perpendicular to the line through the points (-3, 5) and (2, 1) is:

y frac{5}{4}x frac{17}{4}

Alternative and Correct Solution

The alternative solution provided involves using the normal vector of the given line and substituting into the Cartesian equation. The normal vector to the line with points (-3, 5) and (2, 1) is found as follows:

(2 - (-3), 1 - 5) (5, -4)

Using the point (-1, 3) and the normal vector (5, -4), we get:

5x - 4y 5(-1) - 4(3) -5 - 12 -17

Therefore, the Cartesian form of the perpendicular line is:

5x - 4y -17

Conclusion

Both methods provide correct solutions. The slope-intercept form is more common and directly gives the y-intercept. The Cartesian form is useful in certain geometric applications. Understanding these techniques is essential for working with lines in mathematics and computer science.