Can Anyone Find the Equation of a Line Passing Through a Given Point and Perpendicular to Another Line?

In this article, we will walk through the process of determining the equation of a line that passes through a specific point and is perpendicular to another line. This process involves several key concepts in algebra, including understanding slopes and using the point-slope form of a line.

Understanding Slopes

Before we dive into the solution, it's essential to understand what slope means. The slope of a line is a measure of its steepness and is defined as the change in y divided by the change in x. It can be expressed as:

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

Example: Finding the Slope

Let's consider a line that passes through the points (-2, 1) and (1, 2). To find the slope:

Substitute the given points into the formula: [ m frac{2 - 1}{1 - (-2)} frac{1}{3} ]

Determining the Perpendicular Slope

A line that is perpendicular to another line has a slope that is the negative reciprocal of the original line's slope. If the original slope is (frac{1}{3}), then the slope of the perpendicular line is:

[ m_{text{perpendicular}} -frac{1}{frac{1}{3}} -3 ]

Using the Point-Slope Form

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

[ y - y_1 m(x - x_1) ]

Example: Finding the Equation of the Perpendicular Line

We need to find the equation of a line that passes through the point (5, 3) and is perpendicular to the line through points (-2, 1) and (1, 2).

First, find the slope of the given line: [ m frac{1}{3} ] Then, find the perpendicular slope: [ m_{text{perpendicular}} -3 ] Use the point-slope form with (5, 3) and -3: [ y - 3 -3(x - 5) ]

Now, simplify the equation:

Distribute the slope: [ y - 3 -3x 15 ] Add 3 to both sides: [ y -3x 18 ]

Thus, the equation of the line is:

[ y -3x 18 ]

Additional Confusion and Clarification

Some individuals may have trouble with the simplification step. Here is a more detailed explanation of the simplification:

Starting with the point-slope form: [ y - 3 -3(x - 5) ] Distribute the slope: [ y - 3 -3x 15 ] Add 3 to both sides to solve for y: [ y -3x 15 3 ] [ y -3x 18 ]

Verification and Final Equation

The given equation for the line passing through (-2, 1) and (1, 2) can be simplified as:

[ x - frac{1}{1 - (-2)} y - 2] [ x - frac{1}{3} y - 2 ]

Multiplying through by 3 to clear the fraction:

[ 3x - 1 3y - 6 ]

Shifting terms to isolate the line:

[ 3x - 3y -5 ]

To find the line perpendicular to this and passing through (5, 3), we reverse the roles:

[ 3x y 18 ]

Conclusion

Understanding how to find the equation of a line that is both passing through a specific point and perpendicular to another line is a fundamental skill in algebra. Whether you are a student or a professional, mastering these concepts can enhance your problem-solving abilities in mathematics and related fields.