Which Is An Equation For A Line That Is Perpendicular To The Graph Of The Line $y = 2x + 3$?A. $y = 2x + 3$ B. $y = -2x - 3$ C. $y = \frac{1}{2}x + 3$ D. $y = -\frac{1}{2}x - 3$

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Introduction

In mathematics, particularly in geometry and algebra, the concept of perpendicular lines plays a crucial role in understanding various mathematical relationships. When dealing with lines, it's essential to determine the equation of a line that is perpendicular to a given line. In this article, we will explore the concept of perpendicular lines and provide a step-by-step guide on how to find the equation of a line perpendicular to a given line.

Understanding Perpendicular Lines

Perpendicular lines are lines that intersect at a 90-degree angle. In other words, if two lines are perpendicular, they form a right angle at the point of intersection. The concept of perpendicular lines is fundamental in geometry and is used to solve various problems in mathematics.

Slope of a Line

The slope of a line is a measure of how steep the line is. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line is denoted by the letter 'm'. For a line with a positive slope, the line slopes upward from left to right, while for a line with a negative slope, the line slopes downward from left to right.

Finding the Equation of a Line Perpendicular to a Given Line

To find the equation of a line perpendicular to a given line, we need to follow these steps:

  1. Determine the Slope of the Given Line: The first step is to determine the slope of the given line. In this case, the given line is y=2x+3y = 2x + 3. The slope of this line is 2.
  2. Determine the Slope of the Perpendicular Line: The slope of the perpendicular line is the negative reciprocal of the slope of the given line. In this case, the negative reciprocal of 2 is -1/2.
  3. Use the Point-Slope Form: The point-slope form of a line is given by the equation y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the line and (x1,y1)(x_1, y_1) is a point on the line. We can use this form to find the equation of the perpendicular line.
  4. Substitute the Values: Substitute the values of the slope and the point into the point-slope form to get the equation of the perpendicular line.

Finding the Equation of a Line Perpendicular to the Graph of the Line y=2x+3y = 2x + 3

Now, let's apply the steps outlined above to find the equation of a line perpendicular to the graph of the line y=2x+3y = 2x + 3.

  1. Determine the Slope of the Given Line: The slope of the given line is 2.
  2. Determine the Slope of the Perpendicular Line: The slope of the perpendicular line is the negative reciprocal of 2, which is -1/2.
  3. Use the Point-Slope Form: We can use the point-slope form to find the equation of the perpendicular line. Let's choose the point (0, 3) on the given line.
  4. Substitute the Values: Substitute the values of the slope and the point into the point-slope form to get the equation of the perpendicular line.

The equation of the perpendicular line is y−3=−12(x−0)y - 3 = -\frac{1}{2}(x - 0).

Simplifying the Equation

Simplify the equation by multiplying both sides by 2 to get rid of the fraction.

2(y−3)=−12(x−0)2(y - 3) = -\frac{1}{2}(x - 0)

2y−6=−x2y - 6 = -x

2y=−x+62y = -x + 6

y=−12x+3y = -\frac{1}{2}x + 3

Conclusion

In this article, we explored the concept of perpendicular lines and provided a step-by-step guide on how to find the equation of a line perpendicular to a given line. We applied the steps outlined above to find the equation of a line perpendicular to the graph of the line y=2x+3y = 2x + 3. The equation of the perpendicular line is y=−12x+3y = -\frac{1}{2}x + 3.

Introduction

In our previous article, we explored the concept of perpendicular lines and provided a step-by-step guide on how to find the equation of a line perpendicular to a given line. In this article, we will answer some frequently asked questions related to perpendicular lines.

Q&A

Q: What is the difference between perpendicular lines and parallel lines?

A: Perpendicular lines are lines that intersect at a 90-degree angle, while parallel lines are lines that never intersect and are always the same distance apart.

Q: How do you find the equation of a line perpendicular to a given line?

A: To find the equation of a line perpendicular to a given line, you need to determine the slope of the given line, determine the slope of the perpendicular line (which is the negative reciprocal of the slope of the given line), and use the point-slope form to find the equation of the perpendicular line.

Q: What is the negative reciprocal of a slope?

A: The negative reciprocal of a slope is the reciprocal of the slope with a negative sign. For example, the negative reciprocal of 2 is -1/2.

Q: How do you determine the slope of a line?

A: The slope of a line is determined by the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is given by the equation y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the line and (x1,y1)(x_1, y_1) is a point on the line.

Q: Can two lines be both perpendicular and parallel?

A: No, two lines cannot be both perpendicular and parallel. If two lines are perpendicular, they intersect at a 90-degree angle, while if two lines are parallel, they never intersect.

Q: How do you find the equation of a line that is both perpendicular and parallel to a given line?

A: It is not possible to find the equation of a line that is both perpendicular and parallel to a given line, as these two conditions are mutually exclusive.

Q: What is the relationship between the slopes of two perpendicular lines?

A: The slopes of two perpendicular lines are negative reciprocals of each other.

Q: Can a line be perpendicular to itself?

A: No, a line cannot be perpendicular to itself, as a line is a single line and cannot intersect itself at a 90-degree angle.

Q: How do you determine if two lines are perpendicular?

A: To determine if two lines are perpendicular, you can use the slope formula to find the slopes of the two lines and then check if the slopes are negative reciprocals of each other.

Conclusion

In this article, we answered some frequently asked questions related to perpendicular lines. We hope that this article has provided you with a better understanding of the concept of perpendicular lines and how to find the equation of a line perpendicular to a given line.