Line { GH$}$ Passes Through Points { (2,5)$}$ And { (6,9)$}$. Which Equation Represents Line { GH$}$?A. { Y = X + 3$}$B. { Y = X - 3$}$C. { Y = 3x + 3$}$D. { Y = 3x - 3$}$

Mar 9, 2025 by ADMIN 172 views

**Solving Linear Equations: Finding the Equation of a Line**

Introduction

In mathematics, a line is a set of points that extend infinitely in two directions. The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of any point on the line. In this article, we will explore how to find the equation of a line that passes through two given points.

The Problem

We are given two points, (2,5) and (6,9), and we need to find the equation of the line that passes through these two points. The equation of the line is in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

Finding the Slope

The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this case, we can calculate the slope as follows:

m = \frac{y_2 - y_1}{x_2 - x_1}

where (x1, y1) = (2, 5) and (x2, y2) = (6, 9).

m = \frac{9 - 5}{6 - 2}
m = \frac{4}{4}
m = 1

Finding the Equation

Now that we have the slope, we can use either of the two points to find the equation of the line. Let's use the point (2, 5).

y = mx + b
5 = 1(2) + b
5 = 2 + b
b = 3

So, the equation of the line is y = x + 3.

Conclusion

In this article, we have seen how to find the equation of a line that passes through two given points. We calculated the slope of the line using the formula m = (y2 - y1) / (x2 - x1) and then used one of the points to find the equation of the line. The equation of the line is in the form y = mx + b, where m is the slope and b is the y-intercept.

Answer

The correct answer is A. y = x + 3.

Discussion

This problem is a classic example of how to find the equation of a line that passes through two given points. The key concept here is the slope of the line, which is calculated as the ratio of the vertical change to the horizontal change between two points on the line. Once we have the slope, we can use either of the two points to find the equation of the line.

Tips and Variations

To find the equation of a line that passes through three given points, you can use the concept of the slope of the line and the midpoint formula.
To find the equation of a line that passes through a given point and has a given slope, you can use the point-slope form of the equation of a line.
To find the equation of a line that passes through two given points and has a given slope, you can use the slope-intercept form of the equation of a line.

Real-World Applications

In physics, the equation of a line is used to describe the motion of an object under constant acceleration.
In engineering, the equation of a line is used to design and optimize systems such as bridges, buildings, and roads.
In computer science, the equation of a line is used in computer graphics and game development to create realistic and interactive 3D models.

Conclusion

Introduction

In our previous article, we explored how to find the equation of a line that passes through two given points. We calculated the slope of the line using the formula m = (y2 - y1) / (x2 - x1) and then used one of the points to find the equation of the line. In this article, we will answer some frequently asked questions related to finding the equation of a line.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line 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 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 equation of a line?

A: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of any point on the line. It is 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 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 can calculate the slope of the line using the formula m = (y2 - y1) / (x2 - x1) and then use one of the points to find the equation of 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 use the equation y = mx + b and substitute x = 0 into the equation.

Q: What is the point-slope form of the equation of a line?

A: The point-slope form of the equation of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form to find the equation of a line, you can substitute the values of x1, y1, and m into the equation y - y1 = m(x - x1) and then simplify the equation.

Q: What is the slope-intercept form of the equation of a line?

A: The slope-intercept form of the equation of a line is y = mx + b, where m is the slope of the line and b is the y-intercept.

Q: How do I use the slope-intercept form to find the equation of a line?

A: To use the slope-intercept form to find the equation of a line, you can substitute the values of m and b into the equation y = mx + b.

Conclusion

In conclusion, finding the equation of a line that passes through two given points is a fundamental concept in mathematics. The key concept here is the slope of the line, which is calculated as the ratio of the vertical change to the horizontal change between two points on the line. Once we have the slope, we can use either of the two points to find the equation of the line. This concept has numerous real-world applications in physics, engineering, and computer science.

Tips and Variations

To find the equation of a line that passes through three given points, you can use the concept of the slope of the line and the midpoint formula.
To find the equation of a line that passes through a given point and has a given slope, you can use the point-slope form of the equation of a line.
To find the equation of a line that passes through two given points and has a given slope, you can use the slope-intercept form of the equation of a line.

Real-World Applications

In physics, the equation of a line is used to describe the motion of an object under constant acceleration.
In engineering, the equation of a line is used to design and optimize systems such as bridges, buildings, and roads.
In computer science, the equation of a line is used in computer graphics and game development to create realistic and interactive 3D models.