A Line Passes Through The Point \[$(2,7)\$\] With A Slope Of 3. Choose The Correct Equation Of The Line:A) \[$y = 3x + 1\$\]B) \[$y = 3x - 1\$\]
Introduction
In mathematics, a line is defined by its slope and a point it passes through. Given a slope and a point, we can determine the equation of the line. In this article, we will explore how to choose the correct equation of a line with a given slope and a point.
Understanding Slope and Point
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 slope is denoted by the letter 'm' and is calculated as:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
A point on a line is represented by an ordered pair (x, y), where x is the x-coordinate and y is the y-coordinate.
Choosing the Correct Equation
Given a slope of 3 and a point (2, 7), we need to choose the correct equation of the line. The equation of a line can be written in the form:
y = mx + b
where m is the slope and b is the y-intercept.
To choose the correct equation, we need to substitute the given point (2, 7) into the equation and solve for b.
Option A: y = 3x + 1
Let's substitute the point (2, 7) into the equation y = 3x + 1:
7 = 3(2) + 1
Expanding the equation, we get:
7 = 6 + 1
Simplifying the equation, we get:
7 = 7
This equation is true, so the point (2, 7) lies on the line y = 3x + 1.
Option B: y = 3x - 1
Let's substitute the point (2, 7) into the equation y = 3x - 1:
7 = 3(2) - 1
Expanding the equation, we get:
7 = 6 - 1
Simplifying the equation, we get:
7 = 5
This equation is false, so the point (2, 7) does not lie on the line y = 3x - 1.
Conclusion
Based on the analysis, we can conclude that the correct equation of the line with a slope of 3 and a point (2, 7) is:
y = 3x + 1
This equation satisfies the condition that the point (2, 7) lies on the line.
Why is this important?
Choosing the correct equation of a line is important in mathematics and real-world applications. It is used in various fields such as physics, engineering, and economics to model real-world phenomena. In physics, for example, the equation of a line is used to describe the motion of objects. In engineering, it is used to design and optimize systems. In economics, it is used to model economic growth and development.
Real-World Applications
The concept of choosing the correct equation of a line has many real-world applications. Some examples include:
- Physics: The equation of a line is used to describe the motion of objects. For example, the trajectory of a projectile can be modeled using the equation of a line.
- Engineering: The equation of a line is used to design and optimize systems. For example, the equation of a line can be used to model the flow of fluids in a pipe.
- Economics: The equation of a line is used to model economic growth and development. For example, the equation of a line can be used to model the relationship between GDP and inflation.
Conclusion
In conclusion, choosing the correct equation of a line with a given slope and a point is an important concept in mathematics and real-world applications. It is used in various fields such as physics, engineering, and economics to model real-world phenomena. By understanding how to choose the correct equation of a line, we can better understand and analyze real-world data.
References
- Mathematics: "Linear Equations" by Khan Academy
- Physics: "Motion in One Dimension" by OpenStax
- Engineering: "Fluid Mechanics" by OpenStax
- Economics: "Economic Growth and Development" by OpenStax
A Line with a Given Slope and Point: Choosing the Correct Equation - Q&A ====================================================================
Introduction
In our previous article, we explored how to choose the correct equation of a line with a given slope and a point. In this article, we will answer some frequently asked questions (FAQs) related to this topic.
Q: What is the slope of a line?
A: 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.
Q: How do I calculate the slope of a line?
A: To calculate the slope of a line, you need to know the coordinates of two points on the line. The slope is calculated as:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: What is the equation of a line?
A: The equation of a line can be written in the form:
y = mx + b
where m is the slope and b is the y-intercept.
Q: How do I choose the correct equation of a line?
A: To choose the correct equation of a line, you need to substitute the given point into the equation and solve for b. If the equation is true, then the point lies 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 denoted by the letter 'b' in the equation y = mx + b.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, you need to substitute the x-coordinate of the y-intercept (which is 0) into the equation and solve for y.
Q: What is the significance of the y-intercept?
A: The y-intercept is significant because it represents the point where the line intersects the y-axis. It is also used to determine the equation of a line.
Q: Can a line have a negative slope?
A: Yes, a line can have a negative slope. A negative slope indicates that the line slopes downward from left to right.
Q: Can a line have a zero slope?
A: Yes, a line can have a zero slope. A zero slope indicates that the line is horizontal.
Q: Can a line have a positive slope?
A: Yes, a line can have a positive slope. A positive slope indicates that the line slopes upward from left to right.
Q: What is the relationship between the slope and the y-intercept?
A: The slope and the y-intercept are related in that the slope determines the steepness of the line, while the y-intercept determines the point where the line intersects the y-axis.
Conclusion
In conclusion, choosing the correct equation of a line with a given slope and a point is an important concept in mathematics and real-world applications. By understanding how to choose the correct equation of a line, we can better understand and analyze real-world data.
References
- Mathematics: "Linear Equations" by Khan Academy
- Physics: "Motion in One Dimension" by OpenStax
- Engineering: "Fluid Mechanics" by OpenStax
- Economics: "Economic Growth and Development" by OpenStax