Find The Equation Of The Line That Is Parallel To Y = − 5 X − 9 Y = -5x - 9 Y = − 5 X − 9 And Contains The Point ( − 3 , − 3 (-3, -3 ( − 3 , − 3 ].$y = [?]x + [ ]

$$$$

Introduction

In mathematics, the concept of parallel lines is a fundamental idea in geometry and algebra. 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 focus on finding the equation of a line that is parallel to a given line and passes through a specific point. We will use the slope-intercept form of a line, which is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Slope of the Given Line

The given line is $y = -5x - 9$. To find the slope of this line, we can compare it to the slope-intercept form of a line. In this case, the slope is $-5$. This means that for every unit increase in $x$, the value of $y$ decreases by $5$ units.

Slope of the Parallel Line

Since the line we are looking for is parallel to the given line, it must have the same slope. Therefore, the slope of the parallel line is also $-5$.

Point-Slope Form of a Line

To find the equation of the line that passes through the point $(-3, -3)$, we can use the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the slope.

Finding the Equation of the Parallel Line

Substituting the values of the slope and the given point into the point-slope form of a line, we get:

$$y - (-3) = -5(x - (-3))$$

Simplifying this equation, we get:

$$y + 3 = -5(x + 3)$$

Expanding the right-hand side of the equation, we get:

$$y + 3 = -5x - 15$$

Subtracting $3$ from both sides of the equation, we get:

$$y = -5x - 12$$

Conclusion

In this article, we found the equation of the line that is parallel to $y = -5x - 9$ and contains the point $(-3, -3)$. The equation of the parallel line is $y = -5x - 12$. This equation represents a line with a slope of $-5$ and a y-intercept of $-12$.

Applications of Parallel Lines

Parallel lines have many real-world applications. For example, in architecture, parallel lines are used to design buildings and bridges. In engineering, parallel lines are used to design roads and highways. In art, parallel lines are used to create perspective and depth in paintings and drawings.

Examples of Parallel Lines

Here are a few examples of parallel lines:

• The lines on a ruler are parallel.
• The lines on a grid paper are parallel.
• The lines on a coordinate plane are parallel.
• The lines on a map are parallel.

Exercises

Here are a few exercises to practice finding the equation of a line that is parallel to a given line and contains a specific point:

• Find the equation of the line that is parallel to $y = 2x + 1$ and contains the point $(4, 9)$.
• Find the equation of the line that is parallel to $y = -3x - 2$ and contains the point $(-2, 4)$.
• Find the equation of the line that is parallel to $y = x - 1$ and contains the point $(3, 6)$.

Solutions to Exercises

Here are the solutions to the exercises:

• The equation of the line that is parallel to $y = 2x + 1$ and contains the point $(4, 9)$ is $y = 2x + 7$.
• The equation of the line that is parallel to $y = -3x - 2$ and contains the point $(-2, 4)$ is $y = -3x + 10$.
• The equation of the line that is parallel to $y = x - 1$ and contains the point $(3, 6)$ is $y = x + 5$.

Conclusion

In this article, we found the equation of the line that is parallel to $y = -5x - 9$ and contains the point $(-3, -3)$. We also discussed the concept of parallel lines and their applications in real-world scenarios. We provided examples of parallel lines and exercises to practice finding the equation of a line that is parallel to a given line and contains a specific point.

Introduction

In our previous article, we discussed how to find the equation of a line that is parallel to a given line and contains a specific point. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the slope of a line that is parallel to a given line?

A: The slope of a line that is parallel to a given line is the same as the slope of the given line.

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

A: To find the equation of a line that is parallel to a given line and contains a specific point, you can use the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the slope.

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

A: The point-slope form of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and $m$ is the slope.

Q: How do I find the slope of a line from its equation?

A: To find the slope of a line from its equation, you can compare it to the slope-intercept form of a line, which is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is the difference between a line and a plane?

A: A line is a set of points that extend infinitely in two directions, while a plane is a set of points that extend infinitely in three directions.

Q: Can two lines be parallel and intersect at the same point?

A: No, two lines cannot be parallel and intersect at the same point. If two lines intersect at the same point, they are not parallel.

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

A: To determine if two lines are parallel, you can compare their slopes. If the slopes are equal, the lines are parallel.

Q: What is the equation of a line that is parallel to $y = 2x + 1$ and contains the point $(4, 9)$?

A: The equation of the line that is parallel to $y = 2x + 1$ and contains the point $(4, 9)$ is $y = 2x + 7$.

Q: What is the equation of a line that is parallel to $y = -3x - 2$ and contains the point $(-2, 4)$?

A: The equation of the line that is parallel to $y = -3x - 2$ and contains the point $(-2, 4)$ is $y = -3x + 10$.

Q: What is the equation of a line that is parallel to $y = x - 1$ and contains the point $(3, 6)$?

A: The equation of the line that is parallel to $y = x - 1$ and contains the point $(3, 6)$ is $y = x + 5$.

Q: Can a line be parallel to itself?

A: Yes, a line can be parallel to itself.

Q: What is the equation of a line that is parallel to itself and contains the point $(2, 4)$?

A: The equation of the line that is parallel to itself and contains the point $(2, 4)$ is $y = x + 2$.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of a line that is parallel to a given line and contains a specific point. We also provided examples of parallel lines and their equations.