Which Equation Describes The Line That Passes Through The Point { (-3, -2)$}$ And Is Parallel To The Line ${ 6x + 5y = 6\$} ?A. ${ 6x - 5y = -27\$} B. ${ 6x - 5y = -28\$} C. ${ 6x + 5y = -28\$} D.
Which Equation Describes the Line that Passes Through a Given Point and is Parallel to Another Line?
In mathematics, particularly in the field of algebra and geometry, equations of lines play a crucial role in describing the relationship between variables. When dealing with lines, it's essential to understand the concept of parallel lines and how to find the equation of a line that passes through a given point and is parallel to another line. In this article, we will explore this concept and provide a step-by-step guide on how to determine the equation of a line that satisfies these conditions.
Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. In other words, parallel lines have the same slope but different y-intercepts. The equation of a line can be written in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
Finding the Slope of the Given Line
To find the equation of a line that is parallel to the given line, we first need to determine the slope of the given line. The given line is represented by the equation 6x + 5y = 6. To find the slope, we need to rewrite the equation in the slope-intercept form. We can do this by isolating y on one side of the equation.
6x + 5y = 6
5y = -6x + 6
y = (-6/5)x + 6/5
From the rewritten equation, we can see that the slope of the given line is -6/5.
Finding the Equation of the Parallel Line
Now that we have the slope of the given line, we can find the equation of the parallel line that passes through the point (-3, -2). Since the two lines are parallel, they have the same slope, which is -6/5. We can use the point-slope form of a line to find the equation of the parallel line.
The point-slope form of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
y - (-2) = (-6/5)(x - (-3))
y + 2 = (-6/5)(x + 3)
To simplify the equation, we can multiply both sides by 5 to eliminate the fraction.
5(y + 2) = -6(x + 3)
5y + 10 = -6x - 18
6x + 5y = -28
Therefore, the equation of the line that passes through the point (-3, -2) and is parallel to the given line is 6x + 5y = -28.
In conclusion, we have seen how to find the equation of a line that passes through a given point and is parallel to another line. We first determined the slope of the given line by rewriting its equation in the slope-intercept form. Then, we used the point-slope form of a line to find the equation of the parallel line. The final equation of the line that satisfies the given conditions is 6x + 5y = -28.
The equation of a line that passes through a given point and is parallel to another line can be found using the point-slope form of a line. The slope of the given line is determined by rewriting its equation in the slope-intercept form. The point-slope form of a line is then used to find the equation of the parallel line. The final equation of the line that satisfies the given conditions is 6x + 5y = -28.
The correct answer is C. 6x + 5y = -28.
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In our previous article, we discussed how to find the equation of a line that passes through a given point and is parallel to another line. We also provided a step-by-step guide on how to determine the equation of a line that satisfies these conditions. In this article, we will answer some frequently asked questions related to equations of lines and parallel lines.
Q: What is the difference between a line and a parallel line?
A: A line is a set of points that extend infinitely in two directions, while 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.
Q: How do I find the equation of a line that passes through a given point and is parallel to another line?
A: To find the equation of a line that passes through a given point and is parallel to another line, you need to determine the slope of the given line by rewriting its equation in the slope-intercept form. Then, you can use the point-slope form of a line to find the equation of the parallel line.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
Q: How do I rewrite the equation of a line in the slope-intercept form?
A: To rewrite the equation of a line in the slope-intercept form, you need to isolate y on one side of the equation. This can be done by subtracting the x-term from both sides of the equation and then dividing both sides by the coefficient of y.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line is. It is calculated by dividing the change in y by the change in x.
Q: How do I find the equation of a line that is parallel to another line?
A: To find the equation of a line that is parallel to another line, you need to determine the slope of the given line by rewriting its equation in the slope-intercept form. Then, you can use the point-slope form of a line to find the equation of the parallel line.
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 flat surface that extends infinitely in all directions.
Q: How do I find the equation of a line that passes through a given point and is parallel to another line in 3D space?
A: To find the equation of a line that passes through a given point and is parallel to another line in 3D space, you need to determine the direction vector of the line by subtracting the coordinates of the two points. Then, you can use the point-slope form of a line to find the equation of the parallel line.
In conclusion, we have answered some frequently asked questions related to equations of lines and parallel lines. We hope that this article has provided you with a better understanding of these concepts and how to apply them in different situations.
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