Use Point-slope Form To Write The Equation Of A Line That Passes Through The Point \[$(14,1)\$\] With Slope \[$\frac{2}{3}\$\].Answer: \[$\square\$\]
Introduction
In mathematics, the point-slope form is a method used to write the equation of a line that passes through a given point and has a specified slope. This form is particularly useful when we know the coordinates of a point on the line and the slope of the line. In this article, we will explore how to use the point-slope form to write the equation of a line that passes through the point (14, 1) with a slope of 2/3.
What is Point-Slope Form?
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 called the point-slope form because it involves the coordinates of a point on the line and the slope of the line.
How to Use Point-Slope Form
To use the point-slope form to write the equation of a line, we need to know the coordinates of a point on the line and the slope of the line. In this case, we are given the point (14, 1) and the slope 2/3. We can plug these values into the point-slope form to get:
y - 1 = (2/3)(x - 14)
Simplifying the Equation
To simplify the equation, we can start by multiplying both sides of the equation by 3 to eliminate the fraction:
3(y - 1) = 2(x - 14)
Next, we can distribute the 3 to the terms inside the parentheses:
3y - 3 = 2x - 28
Getting the Equation into Slope-Intercept Form
To get the equation into slope-intercept form, we need to isolate y on one side of the equation. We can do this by adding 3 to both sides of the equation:
3y = 2x - 25
Next, we can divide both sides of the equation by 3 to get:
y = (2/3)x - 25/3
Conclusion
In this article, we used the point-slope form to write the equation of a line that passes through the point (14, 1) with a slope of 2/3. We started by plugging the values into the point-slope form and then simplified the equation to get it into slope-intercept form. The final equation is y = (2/3)x - 25/3.
Example Problems
Problem 1
Use the point-slope form to write the equation of a line that passes through the point (2, 3) with a slope of 1/2.
Solution
y - 3 = (1/2)(x - 2)
3(y - 3) = x - 2
3y - 9 = x - 2
3y = x + 7
y = (1/3)x + 7/3
Problem 2
Use the point-slope form to write the equation of a line that passes through the point (5, 2) with a slope of 3.
Solution
y - 2 = 3(x - 5)
3(y - 2) = 3(x - 5)
3y - 6 = 3x - 15
3y = 3x - 9
y = x - 3
Applications of Point-Slope Form
The point-slope form has many applications in mathematics and real-world problems. Some of the applications include:
- Graphing lines: The point-slope form can be used to graph lines by plugging in values for x and y.
- 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 equations: The point-slope form can be used to solve systems of equations by substituting the equation of one line into the equation of another line.
- Modeling real-world problems: The point-slope form can be used to model real-world problems by representing the relationship between two variables as a linear equation.
Conclusion
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 use the point-slope form to write the equation of a line?
A: To use the point-slope form to write the equation of a line, you need to know the coordinates of a point on the line and the slope of the line. You can plug these values into the point-slope form to get:
y - y1 = m(x - x1)
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is given by the equation:
y = mx + b
where m is the slope of the line 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 isolate y on one side of the equation. You can do this by adding y1 to both sides of the equation and then dividing both sides by m.
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 values for x1 and y1
- Not using the correct value for m
- Not simplifying the equation correctly
- Not converting the equation to the slope-intercept form correctly
Q: How do I use the point-slope form to solve systems of equations?
A: To use the point-slope form to solve systems of equations, you need to substitute the equation of one line into the equation of another line. You can then solve for the variables.
Q: What are some real-world applications of the point-slope form?
A: Some real-world applications of the point-slope form include:
- Graphing lines: The point-slope form can be used to graph lines by plugging in values for x and y.
- 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 equations: The point-slope form can be used to solve systems of equations by substituting the equation of one line into the equation of another line.
- Modeling real-world problems: The point-slope form can be used to model real-world problems by representing the relationship between two variables as a linear equation.
Q: How do I choose the correct form of a line?
A: To choose the correct form of a line, you need to consider the information you have about the line. If you know the coordinates of a point on the line and the slope of the line, you can use the point-slope form. If you know the slope and the y-intercept, you can use the slope-intercept form.
Q: What are some tips for using the point-slope form?
A: Some tips for using the point-slope form include:
- Read the problem carefully: Make sure you understand what the problem is asking for.
- Use the correct values: Make sure you use the correct values for x1, y1, and m.
- Simplify the equation: Make sure you simplify the equation correctly.
- Convert to the slope-intercept form: Make sure you convert the equation to the slope-intercept form correctly.
Conclusion
In conclusion, the point-slope form is a powerful tool for writing the equation of a line that passes through a given point and has a specified slope. By using the point-slope form, you can simplify the equation and get it into slope-intercept form. The point-slope form has many applications in mathematics and real-world problems, and it is an essential concept to understand in algebra and geometry.