Write The Point-slope Form Of The Line's Equation Satisfying The Given Conditions. Then Use The Point-slope Form To Write The Slope-intercept Form Of The Equation.Given:- Slope: M = − 3 M = -3 M = − 3 - Passing Through The Point: $\left(-3,
Introduction
In mathematics, the point-slope form of a line's equation is a powerful tool used to describe the relationship between the slope and a point on the line. Given the slope and a point through which the line passes, we can use the point-slope form to write the equation of the line. In this article, we will explore how to write the point-slope form of a line's equation satisfying the given conditions and then use the point-slope form to write the slope-intercept form of the equation.
Understanding the Point-Slope Form
The point-slope form of a line's equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line. This form is called the point-slope form because it uses the slope and a point to write the equation of the line.
Given Conditions
In this problem, we are given the slope (m = -3) and a point (-3, 4) through which the line passes. We will use these conditions to write the point-slope form of the line's equation.
Writing the Point-Slope Form
Using the given conditions, we can write the point-slope form of the line's equation as:
y - 4 = -3(x - (-3))
Simplifying the equation, we get:
y - 4 = -3(x + 3)
Expanding the equation, we get:
y - 4 = -3x - 9
Adding 4 to both sides of the equation, we get:
y = -3x - 5
Converting to Slope-Intercept Form
The slope-intercept form of a line's equation is given by:
y = mx + b
where m is the slope and b is the y-intercept. We can convert the point-slope form to the slope-intercept form by rearranging the terms.
Converting the Point-Slope Form to Slope-Intercept Form
Using the point-slope form of the line's equation (y = -3x - 5), we can see that the slope (m) is -3 and the y-intercept (b) is -5. Therefore, the slope-intercept form of the equation is:
y = -3x - 5
Conclusion
In this article, we have explored how to write the point-slope form of a line's equation satisfying the given conditions and then use the point-slope form to write the slope-intercept form of the equation. We have used the given conditions (slope = -3 and point = (-3, 4)) to write the point-slope form of the line's equation and then converted it to the slope-intercept form. The final equation in slope-intercept form is y = -3x - 5.
Example Problems
Problem 1
Given the slope (m = 2) and a point (1, 3) through which the line passes, write the point-slope form of the line's equation and then convert it to the slope-intercept form.
Solution
Using the given conditions, we can write the point-slope form of the line's equation as:
y - 3 = 2(x - 1)
Simplifying the equation, we get:
y - 3 = 2x - 2
Adding 3 to both sides of the equation, we get:
y = 2x + 1
Converting the point-slope form to the slope-intercept form, we get:
y = 2x + 1
Problem 2
Given the slope (m = -4) and a point (-2, 1) through which the line passes, write the point-slope form of the line's equation and then convert it to the slope-intercept form.
Solution
Using the given conditions, we can write the point-slope form of the line's equation as:
y - 1 = -4(x - (-2))
Simplifying the equation, we get:
y - 1 = -4(x + 2)
Expanding the equation, we get:
y - 1 = -4x - 8
Adding 1 to both sides of the equation, we get:
y = -4x - 7
Converting the point-slope form to the slope-intercept form, we get:
y = -4x - 7
Final Answer
Introduction
In our previous article, we explored how to write the point-slope form of a line's equation satisfying the given conditions and then use the point-slope form to write the slope-intercept form of the equation. In this article, we will answer some frequently asked questions about the point-slope form of a line's equation.
Q&A
Q: What is the point-slope form of a line's equation?
A: The point-slope form of a line's equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
Q: How do I find the point-slope form of a line's equation?
A: To find the point-slope form of a line's equation, you need to know the slope (m) and a point (x1, y1) through which the line passes. You can then use the formula:
y - y1 = m(x - x1)
to write the point-slope form of the equation.
Q: Can I use the point-slope form to find the slope-intercept form of a line's equation?
A: Yes, you can use the point-slope form to find the slope-intercept form of a line's equation. To do this, you need to rearrange the terms in the point-slope form to get:
y = mx + b
where m is the slope and b is the y-intercept.
Q: What is the slope-intercept form of a line's equation?
A: The slope-intercept form of a line's equation is given by:
y = mx + b
where m is the slope and b is the y-intercept.
Q: How do I convert the point-slope form to the slope-intercept form?
A: To convert the point-slope form to the slope-intercept form, you need to rearrange the terms in the point-slope form to get:
y = mx + b
where m is the slope and b is the y-intercept.
Q: Can I use the point-slope form to find the equation of a line that passes through two points?
A: Yes, you can use the point-slope form to find the equation of a line that passes through two points. To do this, you need to find the slope (m) using the two points and then use the point-slope form to write the equation of the line.
Q: What are some common mistakes to avoid when using the point-slope form?
A: Some common mistakes to avoid when using the point-slope form include:
- Not using the correct formula for the point-slope form
- Not using the correct values for the slope (m) and the point (x1, y1)
- Not rearranging the terms correctly to get the slope-intercept form
Example Problems
Problem 1
Given the slope (m = 2) and a point (1, 3) through which the line passes, write the point-slope form of the line's equation and then convert it to the slope-intercept form.
Solution
Using the given conditions, we can write the point-slope form of the line's equation as:
y - 3 = 2(x - 1)
Simplifying the equation, we get:
y - 3 = 2x - 2
Adding 3 to both sides of the equation, we get:
y = 2x + 1
Converting the point-slope form to the slope-intercept form, we get:
y = 2x + 1
Problem 2
Given the slope (m = -4) and a point (-2, 1) through which the line passes, write the point-slope form of the line's equation and then convert it to the slope-intercept form.
Solution
Using the given conditions, we can write the point-slope form of the line's equation as:
y - 1 = -4(x - (-2))
Simplifying the equation, we get:
y - 1 = -4(x + 2)
Expanding the equation, we get:
y - 1 = -4x - 8
Adding 1 to both sides of the equation, we get:
y = -4x - 7
Converting the point-slope form to the slope-intercept form, we get:
y = -4x - 7
Final Answer
The final answer is that the point-slope form of a line's equation is a powerful tool used to describe the relationship between the slope and a point on the line. By using the point-slope form, you can write the equation of a line that passes through a given point and has a given slope.