Lines Parallel and Perpendicular to Given Lines Through Specific Points

Understanding how to find lines that are parallel or perpendicular to a given line, and passing through a specific point, is an important concept in algebra and geometry. This article will guide you through the process and provide practical examples. We will focus on the equations of lines parallel and perpendicular to the line given by 2x - 3y 9 through the point (4, -1).

Parallel Lines

To find a line that is parallel to a given line, we need to understand that the slopes of parallel lines are equal. The given line is in the standard form Ax By C. For our case, the equation is 2x - 3y 9. The slope of this line is m frac{2}{3}. Thus, the slope of any line parallel to it will also be m frac{2}{3}.

Step 1: Writing the Line Parallel to Given Line Through (4, -1)

The general form of a line parallel to 2x - 3y 9 is 2x - 3y D. Here, we use the point (4, -1) to find the value of D.

2(4) - 3(-1) D

8 3 D

D 11

The equation of the parallel line is: 2x - 3y 11.

Perpendicular Lines

To find a line that is perpendicular to a given line, we need to recognize that the slopes of perpendicular lines are negative reciprocals of each other. The slope of the given line is m frac{2}{3}; thus, the slope of a perpendicular line is m -frac{3}{2}.

Step 1: Writing the Line Perpendicular to Given Line Through (4, -1)

The general form of a line perpendicular to 2x - 3y 9 is 3x 2y E. Using the point (4, -1) to find the value of E, we proceed as follows:

3(4) 2(-1) E

12 - 2 E

E 10

The equation of the perpendicular line is: 3x 2y 10.

Graphical Representation

To visualize these lines, we can plot the line xy 0, 2x - 3y 9, x - 4^2y - 1^2 0.01, and the lines 2x - 3y 11 and 3x 2y 10. The range for both x and y will be from -1 to 7 for x and -2 to 4 for y.

Conclusion

We can see that the point (4, -1) lies on the line 2x - 3y 9. The line parallel to 2x - 3y 9 through (4, -1) is 2x - 3y 11, while the line perpendicular to it through the same point is 3x 2y 10. This article provides a clear step-by-step approach to finding parallel and perpendicular lines through a given point.