Write An Equation Of The Line That Passes Through The Given Points $(-4, 7$\] And $(3, 0$\].The Equation Is $\square$. (Type Your Answer In Slope-intercept Form.)
Introduction
In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. The slope-intercept form of a line is a popular method for representing the equation of a line, as it provides a clear and concise way to express the relationship between the x and y coordinates of points on the line. In this article, we will explore how to find the equation of a line that passes through two given points.
What is Slope-Intercept Form?
The slope-intercept form of a line is a mathematical equation that is written in the form:
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 the line is, and 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 of a line is the point where the line intersects the y-axis.
Finding the Slope of a Line
To find the equation of a line that passes through two given points, we need to first find the slope of the line. The slope of a line can be calculated using the following formula:
m = (y2 - y1) / (x2 - x1)
where:
- m is the slope of the line
- (x1, y1) is the first point on the line
- (x2, y2) is the second point on the line
Example: Finding the Slope of a Line
Let's say we have two points on a line: (-4, 7) and (3, 0). We can use the formula above to find the slope of the line.
m = (0 - 7) / (3 - (-4)) m = -7 / 7 m = -1
Finding the Equation of a Line
Now that we have found the slope of the line, we can use the slope-intercept form of a line to find the equation of the line. We can plug in the slope and one of the points on the line to find the value of b, the y-intercept.
Let's say we use the point (-4, 7) to find the value of b.
7 = -1(-4) + b 7 = 4 + b b = 3
The Equation of the Line
Now that we have found the value of b, we can write the equation of the line in slope-intercept form.
y = -x + 3
Conclusion
In this article, we have explored how to find the equation of a line that passes through two given points. We have used the slope-intercept form of a line to find the equation of the line, and we have calculated the slope and y-intercept of the line using the given points. The equation of the line is y = -x + 3.
Tips and Tricks
- Make sure to use the correct formula to find the slope of a line.
- Use the slope-intercept form of a line to find the equation of a line.
- Plug in the slope and one of the points on the line to find the value of b, the y-intercept.
Common Mistakes
- Failing to use the correct formula to find the slope of a line.
- Not using the slope-intercept form of a line to find the equation of a line.
- Not plugging in the slope and one of the points on the line to find the value of b, the y-intercept.
Real-World Applications
The equation of a line has many real-world applications, including:
- Calculating the cost of goods sold
- Determining the amount of interest paid on a loan
- Finding the area of a rectangle
- Calculating the volume of a cylinder
Conclusion
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is a mathematical equation that is written in the form:
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
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the following formula:
m = (y2 - y1) / (x2 - x1)
where:
- m is the slope of the line
- (x1, y1) is the first point on the line
- (x2, y2) is the second point 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 equation of a line that passes through two given points?
A: To find the equation of a line that passes through two given points, you need to first find the slope of the line. Then, you can use the slope-intercept form of a line to find the equation of the line. You can plug in the slope and one of the points on the line to find the value of b, 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:
- Failing to use the correct formula to find the slope of a line
- Not using the slope-intercept form of a line to find the equation of a line
- Not plugging in the slope and one of the points on the line to find the value of b, the y-intercept
Q: What are some real-world applications of the equation of a line?
A: The equation of a line has many real-world applications, including:
- Calculating the cost of goods sold
- Determining the amount of interest paid on a loan
- Finding the area of a rectangle
- Calculating the volume of a cylinder
Q: How do I graph a line using its equation?
A: To graph a line using its equation, you can use the following steps:
- Identify the slope and y-intercept of the line
- Plot the y-intercept on the graph
- Use the slope to determine the direction of the line
- Plot additional points on the line using the slope and y-intercept
Q: What is the difference between a linear equation and a non-linear equation?
A: A linear equation is an equation that can be written in the form:
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
A non-linear equation is an equation that cannot be written in the form above. It may have a quadratic or cubic term, or it may be a transcendental equation.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you can use the following steps:
- Write the equations in the form:
y = mx + b
- Use the substitution method to solve for one variable
- Use the elimination method to solve for the other variable
Q: What are some common types of linear equations?
A: Some common types of linear equations include:
- Slope-intercept form: y = mx + b
- Point-slope form: y - y1 = m(x - x1)
- Standard form: Ax + By = C
Q: How do I use linear equations in real-world applications?
A: Linear equations have many real-world applications, including:
- Calculating the cost of goods sold
- Determining the amount of interest paid on a loan
- Finding the area of a rectangle
- Calculating the volume of a cylinder
Conclusion
In conclusion, finding the equation of a line is a fundamental concept in mathematics. We have explored the slope-intercept form of a line, how to find the slope and y-intercept of a line, and how to use linear equations in real-world applications.