What Is The Equation In Slope-intercept Form Of The Line That Passes Through \[$(-2, 17)\$\] And \[$(3, -13)\$\]?
What is the Equation in Slope-Intercept Form of the Line that Passes Through Two Given Points?
The slope-intercept form of a line is a fundamental concept in mathematics, particularly in algebra and geometry. It is a way to express the equation of a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore how to find the equation in slope-intercept form of the line that passes through two given points.
Understanding the Slope-Intercept Form
The slope-intercept form of a line is given by the equation y = mx + b, where:
- m is the slope of the line
- b is the y-intercept of the line
- x is the independent variable
- y is the dependent variable
The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The y-intercept is the point where the line intersects the y-axis.
Finding the Slope
To find the equation in slope-intercept form of the line that passes through two given points, we need to find the slope of the line first. The slope can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two given points.
Example
Let's consider the two points (-2, 17) and (3, -13). We can use these points to find the slope of the line.
m = (-13 - 17) / (3 - (-2)) m = (-30) / (5) m = -6
Finding the Y-Intercept
Now that we have the slope, we can find the y-intercept by substituting one of the given points into the equation y = mx + b. Let's use the point (-2, 17).
17 = (-6)(-2) + b 17 = 12 + b b = 5
Writing the Equation in Slope-Intercept Form
Now that we have the slope and the y-intercept, we can write the equation in slope-intercept form.
y = -6x + 5
Conclusion
In this article, we have explored how to find the equation in slope-intercept form of the line that passes through two given points. We have used the formula for the slope and the y-intercept to find the equation of the line. The equation in slope-intercept form is a powerful tool for graphing and analyzing lines in mathematics.
Step-by-Step Solution
Here is a step-by-step solution to the problem:
- Find the slope of the line using the formula m = (y2 - y1) / (x2 - x1).
- Substitute one of the given points into the equation y = mx + b to find the y-intercept.
- Write the equation in slope-intercept form using the slope and the y-intercept.
Common Mistakes
Here are some common mistakes to avoid when finding the equation in slope-intercept form:
- Not using the correct formula for the slope.
- Not substituting the correct point into the equation to find the y-intercept.
- Not writing the equation in slope-intercept form correctly.
Real-World Applications
The equation in slope-intercept form has many real-world applications, including:
- Graphing lines on a coordinate plane.
- Analyzing the slope and y-intercept of a line.
- Finding the equation of a line that passes through two given points.
Final Thoughts
In conclusion, finding the equation in slope-intercept form of the line that passes through two given points is a fundamental concept in mathematics. By following the steps outlined in this article, you can find the equation of a line and analyze its slope and y-intercept.
What is the Equation in Slope-Intercept Form of the Line that Passes Through Two Given Points? - Q&A
In our previous article, we explored how to find the equation in slope-intercept form of the line that passes through two given points. In this article, we will answer some of the most frequently asked questions about finding the equation in slope-intercept form.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is a way to express the equation of a line in the form y = mx + b, where m is the slope of the line and 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) and (x2, y2) are two points on the line.
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 given points into the equation y = mx + b and solve for b.
Q: What is the equation in slope-intercept form of a line that passes through the points (2, 3) and (4, 5)?
A: To find the equation in slope-intercept form of a line that passes through the points (2, 3) and (4, 5), you can use the formula m = (y2 - y1) / (x2 - x1) to find the slope, and then substitute one of the points into the equation y = mx + b to find the y-intercept.
m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1
y = 1x + b 3 = 1(2) + b b = 1
y = x + 1
Q: What is the equation in slope-intercept form of a line that passes through the points (-3, 2) and (1, 4)?
A: To find the equation in slope-intercept form of a line that passes through the points (-3, 2) and (1, 4), you can use the formula m = (y2 - y1) / (x2 - x1) to find the slope, and then substitute one of the points into the equation y = mx + b to find the y-intercept.
m = (4 - 2) / (1 - (-3)) m = 2 / 4 m = 1/2
y = (1/2)x + b 2 = (1/2)(-3) + b b = 4
y = (1/2)x + 4
Q: How do I graph a line in slope-intercept form?
A: To graph a line in slope-intercept form, you can use the equation y = mx + b to find the y-intercept, and then use the slope to find the x-intercept.
Q: What is the x-intercept of a line?
A: The x-intercept of a line is the point where the line intersects the x-axis. It is the value of x when y is equal to 0.
Conclusion
In this article, we have answered some of the most frequently asked questions about finding the equation in slope-intercept form of a line that passes through two given points. We have also provided examples of how to find the equation in slope-intercept form of a line that passes through two given points.
Step-by-Step Solution
Here is a step-by-step solution to the problem:
- Find the slope of the line using the formula m = (y2 - y1) / (x2 - x1).
- Substitute one of the given points into the equation y = mx + b to find the y-intercept.
- Write the equation in slope-intercept form using the slope and the y-intercept.
Common Mistakes
Here are some common mistakes to avoid when finding the equation in slope-intercept form:
- Not using the correct formula for the slope.
- Not substituting the correct point into the equation to find the y-intercept.
- Not writing the equation in slope-intercept form correctly.
Real-World Applications
The equation in slope-intercept form has many real-world applications, including:
- Graphing lines on a coordinate plane.
- Analyzing the slope and y-intercept of a line.
- Finding the equation of a line that passes through two given points.
Final Thoughts
In conclusion, finding the equation in slope-intercept form of a line that passes through two given points is a fundamental concept in mathematics. By following the steps outlined in this article, you can find the equation of a line and analyze its slope and y-intercept.