Write An Equation For The Line, In Standard Form, That Passes Through The Following Points:${ \begin{array}{c} (-5, -1) \ (-7, 3) \end{array} }$
Introduction
In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. The standard form of a line is a way to express the equation of a line in a specific format, which is useful for solving problems and graphing lines. In this article, we will discuss how to write an equation for the line that passes through two given points.
What is the Standard Form of a Line?
The standard form of a line is a way to express the equation of a line in the format:
Ax + By = C
where A, B, and C are constants, and x and y are the variables. This format is useful because it allows us to easily identify the slope and y-intercept of the line.
Finding the Slope of the Line
To find the slope of the line, we need to use the formula:
m = (y2 - y1) / (x2 - x1)
where m is the slope, and (x1, y1) and (x2, y2) are the two given points. In this case, the two points are (-5, -1) and (-7, 3).
Plugging in the Values
Let's plug in the values into the formula:
m = (3 - (-1)) / (-7 - (-5))
m = (3 + 1) / (-7 + 5)
m = 4 / -2
m = -2
So, the slope of the line is -2.
Finding the Equation of the Line
Now that we have the slope, we can use the point-slope form of a line to find the equation of the line. The point-slope form is:
y - y1 = m(x - x1)
where (x1, y1) is one of the given points. Let's use the point (-5, -1).
y - (-1) = -2(x - (-5))
y + 1 = -2(x + 5)
y + 1 = -2x - 10
y = -2x - 11
So, the equation of the line is y = -2x - 11.
Converting to Standard Form
To convert the equation to standard form, we need to multiply both sides of the equation by -1.
-y = 2x + 11
2x + y = -11
So, the equation of the line in standard form is 2x + y = -11.
Conclusion
In this article, we discussed how to write an equation for the line that passes through two given points. We used the slope formula to find the slope of the line, and then used the point-slope form to find the equation of the line. Finally, we converted the equation to standard form. The equation of the line in standard form is 2x + y = -11.
Example Problems
- Find the equation of the line that passes through the points (2, 3) and (4, 5).
- Find the equation of the line that passes through the points (-2, 1) and (-1, 4).
- Find the equation of the line that passes through the points (1, 2) and (3, 4).
Solutions
- The slope of the line is (5 - 3) / (4 - 2) = 2 / 2 = 1. The equation of the line is y - 3 = 1(x - 2). Simplifying, we get y = x + 1.
- The slope of the line is (4 - 1) / (-1 - (-2)) = 3 / 1 = 3. The equation of the line is y - 1 = 3(x - (-2)). Simplifying, we get y = 3x + 7.
- The slope of the line is (4 - 2) / (3 - 1) = 2 / 2 = 1. The equation of the line is y - 2 = 1(x - 1). Simplifying, we get y = x + 1.
Tips and Tricks
- When finding the slope of the line, make sure to use the correct formula.
- When using the point-slope form, make sure to use the correct point.
- When converting to standard form, make sure to multiply both sides of the equation by the correct constant.
Conclusion
Q: What is the standard form of a line?
A: The standard form of a line is a way to express the equation of a line in the format Ax + By = C, where A, B, and C are constants, and x and y are the variables.
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1), where m is the slope, and (x1, y1) and (x2, y2) are the two given points.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is one of the given points, and m is the slope.
Q: How do I convert an equation from point-slope form to standard form?
A: To convert an equation from point-slope form to standard form, you need to multiply both sides of the equation by the correct constant.
Q: What are some common mistakes to avoid when writing an equation for a line?
A: Some common mistakes to avoid when writing an equation for a line include:
- Not using the correct formula for the slope
- Not using the correct point in the point-slope form
- Not multiplying both sides of the equation by the correct constant when converting to standard form
- Not simplifying the equation correctly
Q: How do I check if my equation is correct?
A: To check if your equation is correct, you can plug in the given points into the equation and see if it is true. You can also graph the equation and see if it passes through the given points.
Q: What are some real-world applications of writing an equation for a line?
A: Some real-world applications of writing an equation for a line include:
- Finding the equation of a line that passes through two given points
- Finding the equation of a line that is tangent to a circle
- Finding the equation of a line that is perpendicular to a given line
- Finding the equation of a line that passes through a given point and has a given slope
Q: How do I practice writing an equation for a line?
A: To practice writing an equation for a line, you can try the following:
- Use online resources such as Khan Academy or Mathway to practice writing equations for lines
- Work with a partner or tutor to practice writing equations for lines
- Try writing equations for lines with different slopes and intercepts
- Try writing equations for lines that pass through different points
Q: What are some common types of lines that I should know about?
A: Some common types of lines that you should know about include:
- Horizontal lines: These are lines that have a slope of 0 and are parallel to the x-axis.
- Vertical lines: These are lines that have a slope of infinity and are parallel to the y-axis.
- Parallel lines: These are lines that have the same slope and are not the same line.
- Perpendicular lines: These are lines that have slopes that are negative reciprocals of each other.
Q: How do I find the equation of a line that is parallel to a given line?
A: To find the equation of a line that is parallel to a given line, you can use the fact that parallel lines have the same slope. You can then use the point-slope form to find the equation of the line.
Q: How do I find the equation of a line that is perpendicular to a given line?
A: To find the equation of a line that is perpendicular to a given line, you can use the fact that perpendicular lines have slopes that are negative reciprocals of each other. You can then use the point-slope form to find the equation of the line.
Conclusion
In conclusion, writing an equation for a line is a fundamental concept in mathematics that has many real-world applications. By understanding the standard form of a line, the point-slope form, and how to convert between them, you can solve a wide range of problems. With practice and patience, you can become proficient in writing equations for lines.