Use The Given Conditions To Write An Equation For The Line In The Indicated Form. The Line Passes Through \[$(4,3)\$\] And Is Parallel To The Line Whose Equation Is \[$y = 2x - 6\$\]. Write The Equation In Point-slope Form.A. \[$y -
Introduction
In mathematics, the equation of a line can be expressed in various forms, including the slope-intercept form, point-slope form, and standard form. The point-slope form is a popular choice when we know the coordinates of a point on the line and the slope of the line. In this article, we will use the given conditions to write an equation for the line in the point-slope form.
Given Conditions
The line passes through the point {(4,3)$}$ and is parallel to the line whose equation is {y = 2x - 6$}$. We need to find the equation of the line in point-slope form.
Understanding the Slope
The slope of the line {y = 2x - 6$}$ is 2. Since the line we are looking for is parallel to this line, it must have the same slope, which is 2.
Point-Slope Form
The point-slope form of a line is given by the equation:
{y - y_1 = m(x - x_1)$}$
where {(x_1, y_1)$}$ is a point on the line and {m$}$ is the slope of the line.
Substituting the Given Values
We know that the line passes through the point {(4,3)$}$ and has a slope of 2. Substituting these values into the point-slope form equation, we get:
{y - 3 = 2(x - 4)$}$
Simplifying the Equation
To simplify the equation, we can expand the right-hand side:
{y - 3 = 2x - 8$}$
Adding 3 to Both Sides
To isolate the variable {y$}$, we can add 3 to both sides of the equation:
{y = 2x - 8 + 3$}$
Simplifying Further
Simplifying the right-hand side, we get:
{y = 2x - 5$}$
Conclusion
In this article, we used the given conditions to write an equation for the line in the point-slope form. We started by understanding the slope of the line and then used the point-slope form equation to find the equation of the line. The final equation is {y = 2x - 5$}$.
Example Problems
- Find the equation of the line that passes through the point {(2,4)$}$ and is parallel to the line whose equation is {y = 3x + 2$}$.
- Find the equation of the line that passes through the point {(1,2)$}$ and has a slope of 4.
Answer Key
- {y = 3x + 8$}$
- {y = 4x + 2$}$
Tips and Tricks
- When writing an equation in point-slope form, make sure to use the correct slope and point.
- To simplify the equation, expand the right-hand side and combine like terms.
- To isolate the variable, add or subtract the same value from both sides of the equation.
Frequently Asked Questions (FAQs) about Point-Slope Form ===========================================================
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by the equation:
{y - y_1 = m(x - x_1)$}$
where {(x_1, y_1)$}$ is a point on the line and {m$}$ is the slope of the line.
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the slope-intercept form of a line, which is {y = mx + b$}$. The slope {m$}$ is the coefficient of the {x$}$ term.
Q: What is the difference between the point-slope form and the slope-intercept form?
A: The point-slope form and the slope-intercept form are two different ways to write the equation of a line. The point-slope form uses the slope and a point on the line, while the slope-intercept form uses the slope and the y-intercept.
Q: How do I convert a point-slope form equation to a slope-intercept form equation?
A: To convert a point-slope form equation to a slope-intercept form equation, you can expand the right-hand side and simplify the equation.
Q: Can I use the point-slope form to find the equation of a line that is not parallel to another line?
A: Yes, you can use the point-slope form to find the equation of a line that is not parallel to another line. However, you will need to know the slope of the line.
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 can use the point-slope form and the slope formula.
Q: What is the slope formula?
A: The slope formula is:
{m = \frac{y_2 - y_1}{x_2 - x_1}$}$
where {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are two points on the line.
Q: Can I use the point-slope form to find the equation of a line that is perpendicular to another line?
A: Yes, you can use the point-slope form to find the equation of a line that is perpendicular to another line. However, you will need to know the slope of the other line and the slope of the perpendicular line.
Q: How do I find the equation of a line that is perpendicular to another line?
A: To find the equation of a line that is perpendicular to another line, you can use the slope formula and the fact that the slopes of perpendicular lines are negative reciprocals of each other.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope {m$}$ is .
Q: Can I use the point-slope form to find the equation of a line that passes through a point and has a given slope?
A: Yes, you can use the point-slope form to find the equation of a line that passes through a point and has a given slope.
Q: How do I find the equation of a line that passes through a point and has a given slope?
A: To find the equation of a line that passes through a point and has a given slope, you can use the point-slope form and substitute the given slope and point into the equation.
Conclusion
In this article, we have answered some frequently asked questions about the point-slope form of a line. We have covered topics such as the point-slope form equation, finding the slope of a line, converting between point-slope and slope-intercept form, and finding the equation of a line that passes through two points or has a given slope.