Find The Equation Of The Line Passing Through The Given Pair Of Points. Write The Equation In Slope-intercept Form.Points: { \left(\frac{1}{3}, -2\right), \left(\frac{16}{3}, -5\right)$}$What Is The Slope-intercept Form Of The Equation That
Introduction
In mathematics, finding the equation of a line passing through two given points is a fundamental concept in algebra and geometry. The slope-intercept form of a linear equation is a powerful tool for representing lines in a coordinate plane. In this article, we will explore how to find the equation of a line passing through two given points using the slope-intercept form.
What is the Slope-Intercept Form?
The slope-intercept form of a linear equation is a mathematical expression that represents a line in the coordinate plane. It is written in the form:
y = mx + b
where:
- m is the slope of the line
- x is the independent variable
- y is the dependent variable
- b is the y-intercept
Finding the Slope
To find the equation of a line passing through two points, we need to find the slope of the line. The slope of a line is a measure of how steep it is. It can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where:
- (x1, y1) is the first point
- (x2, y2) is the second point
Calculating the Slope
Using the given points (1/3, -2) and (16/3, -5), we can calculate the slope of the line.
m = (-5 - (-2)) / ((16/3) - (1/3)) m = (-5 + 2) / (15/3) m = -3 / 5 m = -0.6
Finding the Equation
Now that we have the slope, we can find the equation of the line using the slope-intercept form. We need to find the y-intercept (b) of the line. To do this, we can substitute one of the points into the equation and solve for b.
Let's use the point (1/3, -2):
-2 = (-0.6)(1/3) + b -2 = -0.2 + b b = -2 + 0.2 b = -1.8
Writing the Equation
Now that we have the slope (m = -0.6) and the y-intercept (b = -1.8), we can write the equation of the line in slope-intercept form:
y = -0.6x - 1.8
Conclusion
In this article, we learned how to find the equation of a line passing through two given points using the slope-intercept form. We calculated the slope of the line using the formula m = (y2 - y1) / (x2 - x1) and found the y-intercept by substituting one of the points into the equation. The final equation of the line is y = -0.6x - 1.8.
Example Problems
- Find the equation of the line passing through the points (2, 3) and (4, 5).
- Find the equation of the line passing through the points (-1, 2) and (3, -4).
Step-by-Step Solutions
- Find the slope:
m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1
Find the y-intercept:
3 = (1)(2) + b 3 = 2 + b b = 1
Write the equation:
y = x + 1
- Find the slope:
m = (-4 - 2) / (3 - (-1)) m = -6 / 4 m = -1.5
Find the y-intercept:
2 = (-1.5)(-1) + b 2 = 1.5 + b b = 0.5
Write the equation:
y = -1.5x + 0.5
Tips and Tricks
- Make sure to calculate the slope correctly using the formula m = (y2 - y1) / (x2 - x1).
- Use a point to find the y-intercept by substituting the values into the equation.
- Write the equation in slope-intercept form (y = mx + b) to represent the line in the coordinate plane.
Common Mistakes
- Calculating the slope incorrectly using the formula m = (y2 - y1) / (x2 - x1).
- Not finding the y-intercept correctly by substituting a point into the equation.
- Writing the equation in the wrong form (e.g., y = mx + b) instead of slope-intercept form.
Conclusion
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is a mathematical expression that represents a line in the coordinate plane. It is written in the form:
y = mx + b
where:
- m is the slope of the line
- x is the independent variable
- y is the dependent variable
- b is the y-intercept
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:
- (x1, y1) is the first point
- (x2, y2) is the second point
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, you can substitute one of the points into the equation and solve for b.
Q: What is the equation of a line in slope-intercept form?
A: The equation of a line in slope-intercept form is written as:
y = mx + b
where:
- m is the slope of the line
- x is the independent variable
- y is the dependent variable
- b is the y-intercept
Q: How do I write the equation of a line in slope-intercept form?
A: To write the equation of a line in slope-intercept form, you need to find the slope (m) and the y-intercept (b) of the line. You can use the formula m = (y2 - y1) / (x2 - x1) to find the slope and substitute one of the points into the equation to find the y-intercept.
Q: What are some common mistakes to avoid when finding the equation of a line?
A: Some common mistakes to avoid when finding the equation of a line include:
- Calculating the slope incorrectly using the formula m = (y2 - y1) / (x2 - x1)
- Not finding the y-intercept correctly by substituting a point into the equation
- Writing the equation in the wrong form (e.g., y = mx + b) instead of slope-intercept form
Q: How do I check if my answer is correct?
A: To check if your answer is correct, you can substitute the values of x and y into the equation and see if it is true. You can also graph the line and check if it passes through the given points.
Q: What are some real-world applications of finding the equation of a line?
A: Finding the equation of a line has many real-world applications, including:
- Calculating the cost of goods sold
- Determining the slope of a roof
- Finding the height of a building
- Calculating the distance between two points
Q: Can I use a calculator to find the equation of a line?
A: Yes, you can use a calculator to find the equation of a line. Many calculators have built-in functions for finding the slope and y-intercept of a line.
Q: How do I find the equation of a line if I only have two points?
A: To find the equation of a line if you only have two points, you can use the formula m = (y2 - y1) / (x2 - x1) to find the slope and substitute one of the points into the equation to find the y-intercept.
Q: Can I find the equation of a line if I have more than two points?
A: Yes, you can find the equation of a line if you have more than two points. You can use the formula m = (y2 - y1) / (x2 - x1) to find the slope and substitute one of the points into the equation to find the y-intercept.
Q: How do I graph a line if I have the equation?
A: To graph a line if you have the equation, you can use a graphing calculator or graph paper. You can also use online graphing tools or software.
Q: Can I find the equation of a line if I have a table of values?
A: Yes, you can find the equation of a line if you have a table of values. You can use the formula m = (y2 - y1) / (x2 - x1) to find the slope and substitute one of the points into the equation to find the y-intercept.
Q: How do I find the equation of a line if I have a word problem?
A: To find the equation of a line if you have a word problem, you need to identify the variables and the relationship between them. You can then use the formula m = (y2 - y1) / (x2 - x1) to find the slope and substitute one of the points into the equation to find the y-intercept.