Find The Line Perpendicular To Y = 3 X − 8 Y = 3x - 8 Y = 3 X − 8 That Includes The Point ( 6 , 1 (6, 1 ( 6 , 1 ].Use The Formula: Y − Y 1 = M ( X − X 1 Y - Y_1 = M(x - X_1 Y − Y 1 ​ = M ( X − X 1 ​ ] Where M M M Is The Slope Of The Perpendicular Line.

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Introduction

In mathematics, finding the equation of a line that is perpendicular to a given line is a common problem. This involves determining the slope of the perpendicular line and using it to find the equation of the line that passes through a given point. In this article, we will discuss how to find the equation of a line that is perpendicular to the given equation y=3x−8y = 3x - 8 and passes through the point (6,1)(6, 1).

Understanding the Problem

To find the equation of a line that is perpendicular to the given equation, we need to understand the concept of slope. The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of the given line y=3x−8y = 3x - 8 is 3, which means that for every unit increase in x, the value of y increases by 3 units.

Finding the Slope of the Perpendicular Line

The slope of the perpendicular line is the negative reciprocal of the slope of the given line. This means that if the slope of the given line is 3, the slope of the perpendicular line is -1/3. To find the equation of the perpendicular line, we need to use the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the perpendicular line.

Using the Formula to Find the Equation of the Perpendicular Line

Now that we have the slope of the perpendicular line, we can use the formula to find its equation. We are given that the point (6,1)(6, 1) lies on the perpendicular line, so we can substitute these values into the formula. The equation of the perpendicular line is y−1=−13(x−6)y - 1 = -\frac{1}{3}(x - 6).

Simplifying the Equation

To simplify the equation, we can multiply both sides by 3 to eliminate the fraction. This gives us 3(y−1)=−1(x−6)3(y - 1) = -1(x - 6). Expanding the left-hand side, we get 3y−3=−x+63y - 3 = -x + 6. Adding 3 to both sides, we get 3y=−x+93y = -x + 9. Finally, dividing both sides by 3, we get y=−13x+3y = -\frac{1}{3}x + 3.

Conclusion

In this article, we have discussed how to find the equation of a line that is perpendicular to the given equation y=3x−8y = 3x - 8 and passes through the point (6,1)(6, 1). We used the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the perpendicular line, to find the equation of the perpendicular line. The final equation of the perpendicular line is y=−13x+3y = -\frac{1}{3}x + 3.

Example Problems

Problem 1

Find the equation of a line that is perpendicular to the given equation y=2x+1y = 2x + 1 and passes through the point (3,4)(3, 4).

Solution

The slope of the given line is 2, so the slope of the perpendicular line is -1/2. Using the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), we get y−4=−12(x−3)y - 4 = -\frac{1}{2}(x - 3). Simplifying the equation, we get y−4=−12x+32y - 4 = -\frac{1}{2}x + \frac{3}{2}. Adding 4 to both sides, we get y=−12x+112y = -\frac{1}{2}x + \frac{11}{2}.

Problem 2

Find the equation of a line that is perpendicular to the given equation y=x−2y = x - 2 and passes through the point (1,3)(1, 3).

Solution

The slope of the given line is 1, so the slope of the perpendicular line is -1. Using the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), we get y−3=−1(x−1)y - 3 = -1(x - 1). Simplifying the equation, we get y−3=−x+1y - 3 = -x + 1. Adding 3 to both sides, we get y=−x+4y = -x + 4.

Applications of Perpendicular Lines

Perpendicular lines have many applications in mathematics and real-life situations. Some examples include:

  • Geometry: Perpendicular lines are used to find the length of the hypotenuse of a right-angled triangle.
  • Physics: Perpendicular lines are used to describe the motion of objects in two dimensions.
  • Engineering: Perpendicular lines are used to design and build structures such as bridges and buildings.
  • Computer Science: Perpendicular lines are used in computer graphics and game development to create 3D models and animations.

Conclusion

In conclusion, finding the equation of a line that is perpendicular to a given equation is a common problem in mathematics. We have discussed how to use the formula y−y1=m(x−x1)y - y_1 = m(x - x_1) to find the equation of the perpendicular line. The final equation of the perpendicular line is y=−13x+3y = -\frac{1}{3}x + 3. We have also provided example problems and discussed the applications of perpendicular lines in mathematics and real-life situations.

Q: What is the slope of a perpendicular line?

A: The slope of a perpendicular line is the negative reciprocal of the slope of the given line. For example, if the slope of the given line is 3, the slope of the perpendicular line is -1/3.

Q: How do I find the equation of a perpendicular line?

A: To find the equation of a perpendicular line, you need to use the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the perpendicular line. You also need to know the coordinates of a point that lies on the perpendicular line.

Q: What is the formula for finding the equation of a perpendicular line?

A: The formula for finding the equation of a perpendicular line is y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the perpendicular line and (x1,y1)(x_1, y_1) are the coordinates of a point that lies on the perpendicular line.

Q: How do I simplify the equation of a perpendicular line?

A: To simplify the equation of a perpendicular line, you can multiply both sides by a common factor to eliminate any fractions. You can also add or subtract the same value to both sides to isolate the variable.

Q: What are some real-life applications of perpendicular lines?

A: Perpendicular lines have many real-life applications, including:

  • Geometry: Perpendicular lines are used to find the length of the hypotenuse of a right-angled triangle.
  • Physics: Perpendicular lines are used to describe the motion of objects in two dimensions.
  • Engineering: Perpendicular lines are used to design and build structures such as bridges and buildings.
  • Computer Science: Perpendicular lines are used in computer graphics and game development to create 3D models and animations.

Q: Can I find the equation of a perpendicular line if I only know the slope of the given line?

A: Yes, you can find the equation of a perpendicular line if you only know the slope of the given line. You can use the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the perpendicular line, which is the negative reciprocal of the slope of the given line.

Q: How do I find the equation of a perpendicular line if I only know the coordinates of two points?

A: To find the equation of a perpendicular line if you only know the coordinates of two points, you can use the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the perpendicular line, which can be found using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

Q: Can I find the equation of a perpendicular line if the given line is not a straight line?

A: No, you cannot find the equation of a perpendicular line if the given line is not a straight line. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to straight lines.

Q: How do I graph a perpendicular line?

A: To graph a perpendicular line, you can use the equation of the line and plot the points that satisfy the equation. You can also use a graphing calculator or software to graph the line.

Q: Can I find the equation of a perpendicular line if I only know the equation of the given line?

A: Yes, you can find the equation of a perpendicular line if you only know the equation of the given line. You can use the formula y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the perpendicular line, which is the negative reciprocal of the slope of the given line.

Q: How do I find the slope of a perpendicular line if I only know the equation of the given line?

A: To find the slope of a perpendicular line if you only know the equation of the given line, you can use the formula m=−1mm = -\frac{1}{m}, where mm is the slope of the given line.

Q: Can I find the equation of a perpendicular line if the given line is a vertical line?

A: No, you cannot find the equation of a perpendicular line if the given line is a vertical line. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to lines that are not vertical.

Q: How do I find the equation of a perpendicular line if the given line is a horizontal line?

A: Yes, you can find the equation of a perpendicular line if the given line is a horizontal line. The equation of the perpendicular line will be a vertical line with the same y-coordinate as the given line.

Q: Can I find the equation of a perpendicular line if the given line is a circle?

A: No, you cannot find the equation of a perpendicular line if the given line is a circle. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to lines that are not circles.

Q: How do I find the equation of a perpendicular line if the given line is a parabola?

A: Yes, you can find the equation of a perpendicular line if the given line is a parabola. The equation of the perpendicular line will be a line that intersects the parabola at a right angle.

Q: Can I find the equation of a perpendicular line if the given line is a hyperbola?

A: Yes, you can find the equation of a perpendicular line if the given line is a hyperbola. The equation of the perpendicular line will be a line that intersects the hyperbola at a right angle.

Q: How do I find the equation of a perpendicular line if the given line is an ellipse?

A: Yes, you can find the equation of a perpendicular line if the given line is an ellipse. The equation of the perpendicular line will be a line that intersects the ellipse at a right angle.

Q: Can I find the equation of a perpendicular line if the given line is a line segment?

A: No, you cannot find the equation of a perpendicular line if the given line is a line segment. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to lines that are not line segments.

Q: How do I find the equation of a perpendicular line if the given line is a ray?

A: No, you cannot find the equation of a perpendicular line if the given line is a ray. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to lines that are not rays.

Q: Can I find the equation of a perpendicular line if the given line is a line with a slope of infinity?

A: No, you cannot find the equation of a perpendicular line if the given line is a line with a slope of infinity. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to lines that have a finite slope.

Q: How do I find the equation of a perpendicular line if the given line is a line with a slope of zero?

A: Yes, you can find the equation of a perpendicular line if the given line is a line with a slope of zero. The equation of the perpendicular line will be a line that intersects the given line at a right angle.

Q: Can I find the equation of a perpendicular line if the given line is a line with a slope of negative infinity?

A: No, you cannot find the equation of a perpendicular line if the given line is a line with a slope of negative infinity. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to lines that have a finite slope.

Q: How do I find the equation of a perpendicular line if the given line is a line with a slope of positive infinity?

A: No, you cannot find the equation of a perpendicular line if the given line is a line with a slope of positive infinity. Perpendicular lines are defined as lines that intersect at a right angle, and this definition only applies to lines that have a finite slope.

Q: Can I find the equation of a perpendicular line if the given line is a line with a slope of 1?

A: Yes, you can find the equation of a perpendicular line if the given line is a line with a slope of 1. The equation of the perpendicular line will be a line that intersects the given line at a right angle.

Q: How do I find the equation of a perpendicular line if the given line is a line with a slope of -1?

A: Yes, you can find the equation of a perpendicular line if the given line is a line with a slope of -1. The equation of the perpendicular line will be a line that intersects the given line at a right angle.

Q: Can I find the equation of a perpendicular line if the given line is a line with a slope of 2?

A: Yes, you can find the equation of a perpendicular line if the given line is a line with a slope of 2. The equation of the perpendicular line will be a line that intersects the given line at a right angle.

Q: How do I find the equation of a perpendicular line if the given line is a line with a slope of -2?

A: Yes, you can find the equation of a perpendicular line if the given line is a line with a slope of