The Equation Of Line { G $}$ Is { Y = 6x - 7 $}$. Line { H $}$, Which Is Parallel To Line { G $}$, Includes The Point { (1, 1) $}$. What Is The Equation Of Line { H $}$?Write The

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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 the equation of a parallel line, given the equation of a known line and a point on the parallel line.

The Equation of Line g

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

The Equation of Line h

Line { h $}$ is parallel to line { g $}$, which means that it has the same slope as line { g $}$. Since the slope of line { g $}$ is 6, the slope of line { h $}$ is also 6.

Finding the Equation of Line h

We are given that line { h $}$ includes the point { (1, 1) $}$. This means that the coordinates of the point { (1, 1) $}$ satisfy the equation of line { h $}$.

Using the Point-Slope Form

To find the equation of line { h $}$, we can use the point-slope form of a linear equation, which is given by:

{ y - y_1 = m(x - x_1) $}$

where { (x_1, y_1) $}$ is a point on the line and { m $}$ is the slope.

Substituting the Values

We know that the slope { m $}$ is 6 and the point { (x_1, y_1) $}$ is { (1, 1) $}$. Substituting these values into the point-slope form, we get:

{ y - 1 = 6(x - 1) $}$

Simplifying the Equation

To simplify the equation, we can expand the right-hand side and combine like terms:

{ y - 1 = 6x - 6 $}$

{ y = 6x - 5 $}$

Conclusion

In this article, we have found the equation of a parallel line, given the equation of a known line and a point on the parallel line. We used the point-slope form of a linear equation and substituted the values of the slope and the point to find the equation of the parallel line.

The Final Answer

The equation of line { h $}$ is { y = 6x - 5 $}$.

Discussion

The concept of parallel lines is a fundamental idea in geometry, and it has many applications in mathematics and science. In this article, we have seen how to find the equation of a parallel line, given the equation of a known line and a point on the parallel line.

Related Topics

  • Slope-Intercept Form: The slope-intercept form of a linear equation is given by { y = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept.
  • Point-Slope Form: The point-slope form of a linear equation is given by { y - y_1 = m(x - x_1) $}$, where { (x_1, y_1) $}$ is a point on the line and { m $}$ is the slope.
  • Parallel Lines: Two lines are said to be parallel if they lie in the same plane and never intersect, no matter how far they are extended.

References

  • Mathematics: The study of numbers, quantities, and shapes.
  • Geometry: The branch of mathematics that deals with the study of shapes, sizes, and positions of objects.
  • Linear Equations: Equations in which the highest power of the variable is 1.

Keywords

  • Parallel Lines
  • Slope-Intercept Form
  • Point-Slope Form
  • Linear Equations
  • Geometry
  • Mathematics

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Introduction

In our previous article, we discussed the concept of parallel lines and how to find the equation of a parallel line, given the equation of a known line and a point on the parallel line. In this article, we will answer some frequently asked questions (FAQs) about parallel lines.

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

A: Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees).

Q: How do I determine if two lines are parallel?

A: To determine if two lines are parallel, you can use the following methods:

  • Check if the lines have the same slope. If they do, then they are parallel.
  • Check if the lines intersect. If they do not intersect, then they are parallel.
  • Check if the lines are in the same plane. If they are, then they are parallel.

Q: What is the equation of a line that is parallel to the line y = 2x + 3?

A: To find the equation of a line that is parallel to the line y = 2x + 3, we need to find a line that has the same slope as the given line. The slope of the given line is 2, so the equation of the parallel line is y = 2x + b, where b is a constant.

Q: How do I find the equation of a line that passes through a given point and is parallel to a given line?

A: To find the equation of a line that passes through a given point and is parallel to a given line, we can use the point-slope form of a linear equation. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the given line.

Q: What is the relationship between the slopes of parallel lines?

A: The slopes of parallel lines are equal. This means that if two lines are parallel, then their slopes are the same.

Q: Can two lines be parallel if they have different slopes?

A: No, two lines cannot be parallel if they have different slopes. If two lines have different slopes, then they are not parallel.

Q: What is the equation of a line that is parallel to the line x = 2y - 3?

A: To find the equation of a line that is parallel to the line x = 2y - 3, we need to find a line that has the same slope as the given line. The slope of the given line is 2, so the equation of the parallel line is x = 2y + b, where b is a constant.

Q: How do I find the equation of a line that is parallel to a given line and passes through a given point?

A: To find the equation of a line that is parallel to a given line and passes through a given point, we can use the point-slope form of a linear equation. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the given line.

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

A: The slopes of perpendicular lines are negative reciprocals of each other. This means that if two lines are perpendicular, then their slopes are negative reciprocals of each other.

Q: Can two lines be perpendicular if they have the same slope?

A: No, two lines cannot be perpendicular if they have the same slope. If two lines have the same slope, then they are parallel, not perpendicular.

Q: What is the equation of a line that is perpendicular to the line y = 2x + 3?

A: To find the equation of a line that is perpendicular to the line y = 2x + 3, we need to find a line that has a slope that is the negative reciprocal of the slope of the given line. The slope of the given line is 2, so the slope of the perpendicular line is -1/2. The equation of the perpendicular line is y = -1/2x + b, where b is a constant.

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

A: To find the equation of a line that is perpendicular to a given line and passes through a given point, we can use the point-slope form of a linear equation. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the perpendicular line.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about parallel lines. We have discussed the difference between parallel lines and perpendicular lines, how to determine if two lines are parallel, and how to find the equation of a line that is parallel to a given line and passes through a given point.

Related Topics

  • Slope-Intercept Form: The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.
  • Point-Slope Form: The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
  • Parallel Lines: Two lines are said to be parallel if they lie in the same plane and never intersect, no matter how far they are extended.
  • Perpendicular Lines: Two lines are said to be perpendicular if they intersect at a right angle (90 degrees).

References

  • Mathematics: The study of numbers, quantities, and shapes.
  • Geometry: The branch of mathematics that deals with the study of shapes, sizes, and positions of objects.
  • Linear Equations: Equations in which the highest power of the variable is 1.

Keywords

  • Parallel Lines
  • Perpendicular Lines
  • Slope-Intercept Form
  • Point-Slope Form
  • Linear Equations
  • Geometry
  • Mathematics