A Line Passes Through The Point \[$(-6, -9)\$\] And Has A Slope Of \[$\frac{5}{2}\$\].Write An Equation In Point-slope Form For This Line.
Introduction
In mathematics, the point-slope form of a line is a way to express the equation of a line using the coordinates of a point on the line and the slope of the line. This form is particularly useful when we know the slope of the line and a point that lies on it. In this article, we will explore how to write an equation in point-slope form for a line that passes through a given point and has a known slope.
The Point-Slope Form of a Line
The point-slope form of a line is given by the equation:
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 also known as the slope-point form.
Writing an Equation in Point-Slope Form
To write an equation in point-slope form, we need to know the coordinates of a point on the line and the slope of the line. In this case, we are given a point (-6, -9) and a slope of 5/2.
Step 1: Identify the Point and the Slope
The point on the line is (-6, -9) and the slope of the line is 5/2.
Step 2: Plug in the Values into the Point-Slope Form
Now that we have identified the point and the slope, we can plug these values into the point-slope form of the equation:
y - y1 = m(x - x1)
Substituting the values, we get:
y - (-9) = (5/2)(x - (-6))
Step 3: Simplify the Equation
To simplify the equation, we can start by evaluating the expressions inside the parentheses:
y + 9 = (5/2)(x + 6)
Next, we can multiply the slope (5/2) by the expression (x + 6) to get:
y + 9 = (5/2)x + 15
Step 4: Write the Equation in Point-Slope Form
Now that we have simplified the equation, we can write it in point-slope form:
y - (-9) = (5/2)(x - (-6))
This is the equation of the line in point-slope form.
Conclusion
In this article, we have learned how to write an equation in point-slope form for a line that passes through a given point and has a known slope. We have used the point-slope form of a line and have plugged in the values of the point and the slope to get the equation of the line. This form is particularly useful when we know the slope of the line and a point that lies on it.
Example
Let's consider another example. Suppose we have a line that passes through the point (2, 3) and has a slope of 2. We can use the point-slope form to write an equation for this line.
Step 1: Identify the Point and the Slope
The point on the line is (2, 3) and the slope of the line is 2.
Step 2: Plug in the Values into the Point-Slope Form
Now that we have identified the point and the slope, we can plug these values into the point-slope form of the equation:
y - y1 = m(x - x1)
Substituting the values, we get:
y - 3 = 2(x - 2)
Step 3: Simplify the Equation
To simplify the equation, we can start by evaluating the expressions inside the parentheses:
y - 3 = 2x - 4
Next, we can add 3 to both sides of the equation to get:
y = 2x - 1
Step 4: Write the Equation in Point-Slope Form
Now that we have simplified the equation, we can write it in point-slope form:
y - 3 = 2(x - 2)
This is the equation of the line in point-slope form.
Applications of the Point-Slope Form
The point-slope form of a line has many applications in mathematics and science. Some of the applications include:
- Graphing Lines: The point-slope form can be used to graph lines on a coordinate plane.
- Finding the Equation of a Line: The point-slope form can be used to find the equation of a line given a point and the slope.
- Solving Systems of Equations: The point-slope form can be used to solve systems of equations.
Conclusion
Introduction
In our previous article, we learned how to write an equation in point-slope form for a line that passes through a given point and has a known slope. In this article, we will answer some frequently asked questions about writing an equation in 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 - 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 write an equation in point-slope form?
A: To write an equation in point-slope form, you need to know the coordinates of a point on the line and the slope of the line. You can then plug these values into the point-slope form of the equation.
Q: What if I don't know the slope of the line?
A: If you don't know the slope of the line, you can use the slope-intercept form of a line, which is given by the equation:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Q: Can I use the point-slope form to graph a line?
A: Yes, you can use the point-slope form to graph a line. To do this, you need to know the coordinates of a point on the line and the slope of the line. You can then use the point-slope form to find the equation of the line and graph it on a coordinate plane.
Q: How do I find the equation of a line given a point and the slope?
A: To find the equation of a line given a point and the slope, you can use the point-slope form of a line. You can plug the coordinates of the point and the slope into the point-slope form of the equation to get the equation of the line.
Q: Can I use the point-slope form to solve systems of equations?
A: Yes, you can use the point-slope form to solve systems of equations. To do this, you need to know the equations of the lines and the points of intersection. You can then use the point-slope form to find the equation of each line and solve the system of equations.
Q: What are some common mistakes to avoid when writing an equation in point-slope form?
A: Some common mistakes to avoid when writing an equation in point-slope form include:
- Not plugging in the correct values: Make sure to plug in the correct values for the point and the slope.
- Not simplifying the equation: Make sure to simplify the equation to get the final answer.
- Not using the correct form: Make sure to use the correct form of the equation, which is y - y1 = m(x - x1).
Q: How do I check my work when writing an equation in point-slope form?
A: To check your work when writing an equation in point-slope form, you can:
- Plug in the values: Plug in the values of the point and the slope into the point-slope form of the equation.
- Simplify the equation: Simplify the equation to get the final answer.
- Check the equation: Check the equation to make sure it is correct.
Conclusion
In this article, we have answered some frequently asked questions about writing an equation in point-slope form. We have covered topics such as the point-slope form of a line, how to write an equation in point-slope form, and common mistakes to avoid. We hope this article has been helpful in answering your questions about writing an equation in point-slope form.