The Equation Of The Line Containing The Points { (1,3)$}$ And { (-1,2)$}$ Is:A. { Y = 2x + 1$}$B. { Y = -\frac{1}{2}x + \frac{7}{2}$}$C. { Y = \frac{1}{2}x + \frac{5}{2}$}$D. { Y = -2x + 5$}$

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Introduction

In mathematics, the equation of a line is a fundamental concept that describes the relationship between the x and y coordinates of points on a line. Given two points on a line, we can find the equation of the line using various methods, including the slope-intercept form and the point-slope form. In this article, we will explore how to find the equation of a line containing two points, and we will apply this concept to solve a problem involving two given points.

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 and b is the y-intercept. 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.

Finding the Slope

To find the slope of a line containing two points, we can use the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Example: Finding the Slope

Let's consider the two points (1, 3) and (-1, 2). To find the slope of the line containing these points, we can use the formula:

m = (2 - 3) / (-1 - 1) m = -1 / -2 m = 1/2

The Point-Slope Form

The point-slope form of a line is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope of the line.

Finding the Equation of the Line

Now that we have found the slope of the line containing the points (1, 3) and (-1, 2), we can use the point-slope form to find the equation of the line. Let's use the point (1, 3) as the point (x1, y1) in the equation.

y - 3 = (1/2)(x - 1)

To simplify the equation, we can multiply both sides by 2 to eliminate the fraction:

2(y - 3) = x - 1

Expanding the left-hand side of the equation, we get:

2y - 6 = x - 1

Adding 6 to both sides of the equation, we get:

2y = x + 5

Dividing both sides of the equation by 2, we get:

y = (1/2)x + 5/2

Conclusion

In this article, we have explored how to find the equation of a line containing two points. We have used the slope-intercept form and the point-slope form to find the equation of the line, and we have applied this concept to solve a problem involving two given points. The equation of the line containing the points (1, 3) and (-1, 2) is:

y = (1/2)x + 5/2

This equation describes the relationship between the x and y coordinates of points on the line, and it can be used to find the y-coordinate of any point on the line given its x-coordinate.

The Final Answer

The final answer is:

Introduction

In our previous article, we explored how to find the equation of a line containing two points. We used the slope-intercept form and the point-slope form to find the equation of the line, and we applied this concept to solve a problem involving two given points. In this article, we will answer some frequently asked questions about the equation of a 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 points on a line. It is typically written 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 containing two points?

A: To find the equation of a line containing two points, you can use the slope-intercept form or the point-slope form. The slope-intercept form is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the 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. The slope of a line can be positive, negative, or zero.

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 the coordinates of 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 zero.

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, where m is the slope of the line and b is the y-intercept. The y-intercept is the value of b.

Q: Can I find the equation of a line using only one point?

A: No, you cannot find the equation of a line using only one point. You need at least two points to find the equation of a line.

Q: What if the two points are not on the same line?

A: If the two points are not on the same line, then there is no equation of the line that contains both points.

Q: Can I use the equation of a line to find the coordinates of a point?

A: Yes, you can use the equation of a line to find the coordinates of a point. Given the equation of a line and the x-coordinate of a point, you can substitute the x-coordinate into the equation and solve for the y-coordinate.

Conclusion

In this article, we have answered some frequently asked questions about the equation of a line. We have discussed the slope-intercept form and the point-slope form, and we have provided examples of how to find the equation of a line containing two points. We have also discussed the y-intercept of a line and how to find it. We hope that this article has been helpful in answering your questions about the equation of a line.

The Final Answer

The final answer is:

  • The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of points on a line.
  • The slope of a line is a measure of how steep the line is.
  • The y-intercept of a line is the point where the line intersects the y-axis.
  • You need at least two points to find the equation of a line.
  • You can use the equation of a line to find the coordinates of a point.