X 15. The Line Represented By Y 4 + = 1, Intersects X-axis And Y-axis 6 Respectively At P And Q. The Coordinates Of The Mid-point Of Line Segment PQ Are: (A) (2,3) (C) (2,0) (B) (3,2) (D) (0, 3)

Mar 10, 2025 by ADMIN 196 views

Solving Linear Equations: Finding the Midpoint of Line Segment PQ

In this article, we will explore the concept of linear equations and how to find the midpoint of a line segment. We will use the given equation y = 4x + 1 to find the coordinates of the midpoint of line segment PQ, where P and Q are the points of intersection of the line with the x-axis and y-axis respectively.

Understanding the Equation

The given equation is y = 4x + 1. This is a linear equation in the slope-intercept form, where the slope (m) is 4 and the y-intercept (c) is 1.

Finding the x-Intercept

To find the x-intercept, we need to set y = 0 and solve for x.

0 = 4x + 1

Subtracting 1 from both sides:

-1 = 4x

Dividing both sides by 4:

x = -1/4

So, the x-intercept is (-1/4, 0).

Finding the y-Intercept

The y-intercept is already given as (0, 1).

Finding the Midpoint

The midpoint of a line segment is the point that divides the line segment into two equal parts. To find the midpoint, we need to find the average of the x-coordinates and the y-coordinates of the two points.

Let's find the midpoint of line segment PQ.

The x-coordinate of the midpoint is the average of the x-coordinates of P and Q:

x-coordinate of midpoint = (-1/4 + 0)/2 = -1/8

The y-coordinate of the midpoint is the average of the y-coordinates of P and Q:

y-coordinate of midpoint = (0 + 1)/2 = 1/2

So, the coordinates of the midpoint of line segment PQ are (-1/8, 1/2).

In this article, we used the given equation y = 4x + 1 to find the coordinates of the midpoint of line segment PQ. We first found the x-intercept and y-intercept of the line, and then used the midpoint formula to find the coordinates of the midpoint.

The correct answer is (A) (-1/8, 1/2).

The x-intercept of a line is the point where the line intersects the x-axis.
The y-intercept of a line is the point where the line intersects the y-axis.
The midpoint of a line segment is the point that divides the line segment into two equal parts.
To find the midpoint of a line segment, we need to find the average of the x-coordinates and the y-coordinates of the two points.

Find the x-intercept and y-intercept of the line y = 2x - 3.
Find the midpoint of the line segment joining the points (2, 3) and (4, 5).
Find the equation of the line passing through the points (1, 2) and (3, 4).

To find the x-intercept, we need to set y = 0 and solve for x.

0 = 2x - 3

Adding 3 to both sides:

3 = 2x

Dividing both sides by 2:

x = 3/2

So, the x-intercept is (3/2, 0).

To find the y-intercept, we need to set x = 0 and solve for y.

y = 2(0) - 3

y = -3

So, the y-intercept is (0, -3).
To find the midpoint, we need to find the average of the x-coordinates and the y-coordinates of the two points.

x-coordinate of midpoint = (2 + 4)/2 = 3

y-coordinate of midpoint = (3 + 5)/2 = 4

So, the coordinates of the midpoint are (3, 4).
To find the equation of the line, we need to find the slope and the y-intercept.

The slope is (y2 - y1)/(x2 - x1) = (4 - 2)/(3 - 1) = 2/2 = 1

The y-intercept is the point where the line intersects the y-axis. We can find this by substituting x = 0 into the equation.

y = 1(0) + b

y = b

So, the equation of the line is y = x + b. We can find the value of b by substituting the point (1, 2) into the equation.

2 = 1(1) + b

2 = 1 + b

Subtracting 1 from both sides:

1 = b

So, the equation of the line is y = x + 1.
Solving Linear Equations: Q&A

In our previous article, we explored the concept of linear equations and how to find the midpoint of a line segment. We used the given equation y = 4x + 1 to find the coordinates of the midpoint of line segment PQ. In this article, we will answer some frequently asked questions related to linear equations and solving them.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. It can be written in the form y = mx + c, where m is the slope and c is the y-intercept.

Q: How do I find the x-intercept of a line?

A: To find the x-intercept, you need to set y = 0 and solve for x. This will give you the point where the line intersects the x-axis.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept, you need to set x = 0 and solve for y. This will give you the point where the line intersects the y-axis.

Q: What is the midpoint of a line segment?

A: The midpoint of a line segment is the point that divides the line segment into two equal parts. It is the average of the x-coordinates and the y-coordinates of the two points.

Q: How do I find the midpoint of a line segment?

A: To find the midpoint, you need to find the average of the x-coordinates and the y-coordinates of the two points. This can be done using the formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

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 find the slope of a line?

A: To find the slope, you need to use the formula:

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

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 the points on the line. It can be written in the form y = mx + c, where m is the slope and c is the y-intercept.

Q: How do I find the equation of a line?

A: To find the equation of a line, you need to find the slope and the y-intercept. This can be done using the point-slope form of a line, which is:

y - y1 = m(x - x1)

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

A: The point-slope form of a line is a mathematical expression that describes the relationship between the x and y coordinates of the points on the line. It is written in the form:

y - y1 = m(x - x1)

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

A: To use the point-slope form, you need to substitute the values of the slope (m) and the coordinates of a point (x1, y1) into the equation. This will give you the equation of the line.

In this article, we answered some frequently asked questions related to linear equations and solving them. We covered topics such as finding the x-intercept and y-intercept of a line, finding the midpoint of a line segment, and finding the equation of a line. We hope that this article has been helpful in clarifying any doubts you may have had about linear equations.

Find the x-intercept and y-intercept of the line y = 2x - 3.
Find the midpoint of the line segment joining the points (2, 3) and (4, 5).
Find the equation of the line passing through the points (1, 2) and (3, 4).

To find the x-intercept, we need to set y = 0 and solve for x.

0 = 2x - 3

Adding 3 to both sides:

3 = 2x

Dividing both sides by 2:

x = 3/2

So, the x-intercept is (3/2, 0).

To find the y-intercept, we need to set x = 0 and solve for y.

y = 2(0) - 3

y = -3

So, the y-intercept is (0, -3).
To find the midpoint, we need to find the average of the x-coordinates and the y-coordinates of the two points.

x-coordinate of midpoint = (2 + 4)/2 = 3

y-coordinate of midpoint = (3 + 5)/2 = 4

So, the coordinates of the midpoint are (3, 4).
To find the equation of the line, we need to find the slope and the y-intercept.

The slope is (y2 - y1)/(x2 - x1) = (4 - 2)/(3 - 1) = 2/2 = 1

The y-intercept is the point where the line intersects the y-axis. We can find this by substituting x = 0 into the equation.

y = 1(0) + b

y = b

So, the equation of the line is y = x + b. We can find the value of b by substituting the point (1, 2) into the equation.

2 = 1(1) + b

2 = 1 + b

Subtracting 1 from both sides:

1 = b

So, the equation of the line is y = x + 1.