The Equation For Line \[$ G \$\] Can Be Written As $ Y = \frac{7}{2} X + 6 $. Line \[$ H \$\] Includes The Point \[$(-2, -3)\$\] And Is Parallel To Line \[$ G \$\]. What Is The Equation Of Line \[$ H

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Introduction

In mathematics, the concept of parallel lines is a fundamental idea in geometry. Two lines are said to be parallel if they lie in the same plane and never intersect, no matter how far they are extended. In this article, we will explore how to find the equation of a line that is parallel to a given line. We will use the concept of slope and the point-slope form of a linear equation to derive the equation of the parallel line.

The Given Line

The equation of the given line, denoted as line { g $}$, is $ y = \frac{7}{2} x + 6 $. This is a linear equation in the slope-intercept form, where the slope is 72\frac{7}{2} and the y-intercept is 6.

The Parallel Line

Line { h $}$ is parallel to line { g $}$ and includes the point {(-2, -3)$}$. To find the equation of line { h $}$, we need to determine its slope and then use the point-slope form of a linear equation.

Finding the Slope

Since line { h $}$ is parallel to line { g $}$, it has the same slope as line { g $}$. The slope of line { g $}$ is 72\frac{7}{2}, so the slope of line { h $}$ is also 72\frac{7}{2}.

Using the Point-Slope Form

The point-slope form of a linear equation is given by:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope. In this case, we have the point (−2,−3)(-2, -3) and the slope 72\frac{7}{2}.

Substituting these values into the point-slope form, we get:

y−(−3)=72(x−(−2))y - (-3) = \frac{7}{2}(x - (-2))

Simplifying this equation, we get:

y+3=72(x+2)y + 3 = \frac{7}{2}(x + 2)

Simplifying the Equation

To simplify the equation, we can multiply both sides by 2 to eliminate the fraction:

2(y+3)=7(x+2)2(y + 3) = 7(x + 2)

Expanding the left-hand side, we get:

2y+6=7x+142y + 6 = 7x + 14

Subtracting 6 from both sides, we get:

2y=7x+82y = 7x + 8

Dividing both sides by 2, we get:

y=72x+4y = \frac{7}{2}x + 4

Conclusion

In this article, we have derived the equation of a line that is parallel to a given line. We used the concept of slope and the point-slope form of a linear equation to find the equation of the parallel line. The equation of line { h $}$ is y=72x+4y = \frac{7}{2}x + 4. This equation is in the slope-intercept form, where the slope is 72\frac{7}{2} and the y-intercept is 4.

The Importance of Parallel Lines

Parallel lines are an important concept in geometry and are used in many real-world applications, such as architecture, engineering, and computer graphics. Understanding how to find the equation of a parallel line is essential for solving problems in these fields.

Real-World Applications

Parallel lines have many real-world applications, such as:

  • Architecture: Architects use parallel lines to design buildings and structures.
  • Engineering: Engineers use parallel lines to design bridges, roads, and other infrastructure.
  • Computer Graphics: Computer graphics artists use parallel lines to create 3D models and animations.

Conclusion

Introduction

In our previous article, we derived the equation of a line that is parallel to a given line. We used the concept of slope and the point-slope form of a linear equation to find the equation of the parallel line. In this article, we will answer some frequently asked questions about parallel lines and their equations.

Q: What is the difference between a parallel line and a perpendicular line?

A: A parallel line is a line that lies in the same plane as another line and never intersects it, no matter how far they are extended. A perpendicular line, on the other hand, is a line that intersects another line at a right angle (90 degrees).

Q: How do I find the equation of a line that is perpendicular to a given line?

A: To find the equation of a line that is perpendicular to a given line, you need to find the slope of the given line and then take the negative reciprocal of that slope. The negative reciprocal of a slope is a number that, when multiplied by the original slope, gives -1.

Q: What is the slope of a line that is parallel to the x-axis?

A: The slope of a line that is parallel to the x-axis is 0. This is because the x-axis is a horizontal line, and the slope of a horizontal line is always 0.

Q: What is the slope of a line that is parallel to the y-axis?

A: The slope of a line that is parallel to the y-axis is undefined. This is because the y-axis is a vertical line, and the slope of a vertical line is undefined.

Q: Can a line be both parallel and perpendicular to another line?

A: No, a line cannot be both parallel and perpendicular to another line. These two concepts are mutually exclusive, and a line can only be one or the other.

Q: How do I find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you need to use the point-slope form of a linear equation. This involves finding the slope of the line using the two points and then using one of the points to find the equation of the line.

Q: What is the equation of a line that passes through the points (2, 3) and (4, 5)?

A: To find the equation of a line that passes through the points (2, 3) and (4, 5), we need to find the slope of the line using the two points. The slope is given by:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Substituting the values of the two points, we get:

m=5−34−2=22=1m = \frac{5 - 3}{4 - 2} = \frac{2}{2} = 1

Now that we have the slope, we can use one of the points to find the equation of the line. Let's use the point (2, 3). The equation of the line is given by:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Substituting the values of the point and the slope, we get:

y−3=1(x−2)y - 3 = 1(x - 2)

Simplifying this equation, we get:

y−3=x−2y - 3 = x - 2

Adding 3 to both sides, we get:

y=x+1y = x + 1

Conclusion

In this article, we have answered some frequently asked questions about parallel lines and their equations. We have discussed the difference between parallel and perpendicular lines, how to find the equation of a line that is perpendicular to a given line, and how to find the equation of a line that passes through two points. We have also provided examples of how to find the equation of a line that passes through two points.