Write The Point-slope Form Of The Equation For A Line That Passes Through \[$(-1, 4)\$\] With A Slope Of 2.The Value Of \[$x_1\$\] Is \[$\square\$\].The Value Of \[$y_1\$\] Is \[$\square\$\].The Point-slope Form
Introduction
In mathematics, the point-slope form of a line is a fundamental concept that helps us to write the equation of a line that passes through a given point and has a specified slope. In this article, we will explore the point-slope form of a line, its significance, and how to use it to write the equation of a line that passes through a given point with a specified slope.
What is the Point-Slope Form of a Line?
The point-slope form of a line is a mathematical equation that represents a line that passes through a given point and has a specified slope. The point-slope form is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.
Understanding the Components of the Point-Slope Form
To write the point-slope form of a line, we need to understand the components of the equation. The components are:
- m: The slope of the line, which is a measure of how steep the line is.
- (x1, y1): The given point through which the line passes.
- y - y1: The difference between the y-coordinate of any point on the line and the y-coordinate of the given point.
- x - x1: The difference between the x-coordinate of any point on the line and the x-coordinate of the given point.
How to Write the Point-Slope Form of a Line
To write the point-slope form of a line, we need to follow these steps:
- Identify the given point: The given point is (x1, y1).
- Identify the slope: The slope of the line is m.
- Write the equation: Substitute the values of x1, y1, and m into the point-slope form equation.
Example: Writing the Point-Slope Form of a Line
Let's consider an example to illustrate how to write the point-slope form of a line.
Suppose we want to write the point-slope form of a line that passes through the point (-1, 4) with a slope of 2.
Step 1: Identify the given point
The given point is (-1, 4).
Step 2: Identify the slope
The slope of the line is 2.
Step 3: Write the equation
Substitute the values of x1, y1, and m into the point-slope form equation:
y - 4 = 2(x - (-1))
Simplify the equation:
y - 4 = 2(x + 1)
y - 4 = 2x + 2
y = 2x + 6
Therefore, the point-slope form of the equation of the line is y = 2x + 6.
The Value of x1
In the example above, the value of x1 is -1.
The Value of y1
In the example above, the value of y1 is 4.
Conclusion
In conclusion, the point-slope form of a line is a mathematical equation that represents a line that passes through a given point and has a specified slope. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the line. By following the steps outlined in this article, we can write the point-slope form of a line that passes through a given point with a specified slope.
Frequently Asked Questions
Q: What is the point-slope form of a line?
A: The point-slope form of a line is a mathematical equation that represents a line that passes through a given point and has a specified slope.
Q: How do I write the point-slope form of a line?
A: To write the point-slope form of a line, identify the given point, identify the slope, and substitute the values into the point-slope form equation.
Q: What is the significance of the point-slope form of a line?
A: The point-slope form of a line is significant because it helps us to write the equation of a line that passes through a given point and has a specified slope.
Q: How do I use the point-slope form of a line in real-life applications?
A: The point-slope form of a line can be used in real-life applications such as graphing lines, finding the equation of a line, and solving systems of linear equations.
Q: What are some common mistakes to avoid when writing the point-slope form of a line?
A: Some common mistakes to avoid when writing the point-slope form of a line include:
- Not identifying the given point correctly
- Not identifying the slope correctly
- Not substituting the values into the point-slope form equation correctly
By avoiding these common mistakes, we can ensure that we write the point-slope form of a line correctly and accurately.
Frequently Asked Questions: Point-Slope Form of a Line
Q: What is the point-slope form of a line?
A: The point-slope form of a line is a mathematical equation that represents a line that passes through a given point and has a specified slope. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the line.
Q: How do I write the point-slope form of a line?
A: To write the point-slope form of a line, follow these steps:
- Identify the given point: The given point is (x1, y1).
- Identify the slope: The slope of the line is m.
- Write the equation: Substitute the values of x1, y1, and m into the point-slope form equation.
Q: What is the significance of the point-slope form of a line?
A: The point-slope form of a line is significant because it helps us to write the equation of a line that passes through a given point and has a specified slope. This is useful in various applications such as graphing lines, finding the equation of a line, and solving systems of linear equations.
Q: How do I use the point-slope form of a line in real-life applications?
A: The point-slope form of a line can be used in real-life applications such as:
- Graphing lines: The point-slope form can be used to graph lines by plotting the given point and using the slope to determine the direction of the line.
- Finding the equation of a line: The point-slope form can be used to find the equation of a line that passes through a given point and has a specified slope.
- Solving systems of linear equations: The point-slope form can be used to solve systems of linear equations by finding the intersection point of two lines.
Q: What are some common mistakes to avoid when writing the point-slope form of a line?
A: Some common mistakes to avoid when writing the point-slope form of a line include:
- Not identifying the given point correctly: Make sure to identify the given point correctly and substitute the values into the point-slope form equation.
- Not identifying the slope correctly: Make sure to identify the slope correctly and substitute the values into the point-slope form equation.
- Not substituting the values into the point-slope form equation correctly: Make sure to substitute the values into the point-slope form equation correctly to avoid errors.
Q: Can I use the point-slope form of a line to find the equation of a line that passes through two points?
A: Yes, you can use the point-slope form of a line to find the equation of a line that passes through two points. To do this, follow these steps:
- Find the slope: Find the slope of the line using the two points.
- Write the point-slope form equation: Substitute the values of the two points and the slope into the point-slope form equation.
- Simplify the equation: Simplify the equation to find the final equation of the line.
Q: Can I use the point-slope form of a line to solve systems of linear equations?
A: Yes, you can use the point-slope form of a line to solve systems of linear equations. To do this, follow these steps:
- Find the intersection point: Find the intersection point of the two lines by setting the two equations equal to each other.
- Write the point-slope form equation: Substitute the values of the intersection point into the point-slope form equation.
- Simplify the equation: Simplify the equation to find the final solution to the system of linear equations.
Q: What are some real-life applications of the point-slope form of a line?
A: Some real-life applications of the point-slope form of a line include:
- Graphing lines: The point-slope form can be used to graph lines by plotting the given point and using the slope to determine the direction of the line.
- Finding the equation of a line: The point-slope form can be used to find the equation of a line that passes through a given point and has a specified slope.
- Solving systems of linear equations: The point-slope form can be used to solve systems of linear equations by finding the intersection point of two lines.
- Modeling real-world situations: The point-slope form can be used to model real-world situations such as the motion of an object, the growth of a population, and the cost of a product.
Q: How do I determine the slope of a line?
A: To determine the slope of a line, follow these steps:
- Find two points on the line: Find two points on the line that are not the same point.
- Calculate the slope: Calculate the slope using the formula m = (y2 - y1) / (x2 - x1).
- Simplify the equation: Simplify the equation to find the final slope of the line.
Q: How do I determine the equation of a line that passes through a given point and has a specified slope?
A: To determine the equation of a line that passes through a given point and has a specified slope, follow these steps:
- Identify the given point: Identify the given point (x1, y1).
- Identify the slope: Identify the slope m.
- Write the point-slope form equation: Substitute the values of x1, y1, and m into the point-slope form equation.
- Simplify the equation: Simplify the equation to find the final equation of the line.
By following these steps and avoiding common mistakes, you can use the point-slope form of a line to find the equation of a line that passes through a given point and has a specified slope.