Question

Plot each pair of points and determine the slope of the line containing the points. Graph the line. $$ (4,2) ;(-5,2) $$

Short answer

The slope is 0, and the line is horizontal through \[ y = 2 \].

Step-by-step solution

- Plot the Points

Plot the points \(4,2\) and \(-5,2\) on a coordinate plane. These will be the points through which the line passes.

- Determine the Slope

To find the slope \(m\) of the line passing through the two points \( (x_1, y_1) \) and \((x_2, y_2)\), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given coordinates \( (4,2) \) and \$(-5, 2) \, we get: \[ m = \frac{2 - 2}{-5 - 4} = \frac{0}{-9} = 0 \] Therefore, the slope of the line is 0.

- Interpret the Slope

A slope of 0 indicates that the line is horizontal. This means the y-coordinates of all points on the line are the same.

- Graph the Line

Draw a horizontal line passing through the points \(4,2\) and \(-5,2\) on the coordinate plane. This line represents the equation \ y = 2 \.

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal line called the x-axis and a vertical line called the y-axis. These axes intersect at a point known as the origin, labeled as (0,0). Each point on the plane is represented by a pair of numbers \(x, y\), called coordinates.
To plot a point, start at the origin.
Move horizontally to the x-coordinate, and then move vertically to the y-coordinate.

• For the point (4,2): Start at the origin, move 4 units to the right and 2 units up.
• For the point (-5,2): Start at the origin, move 5 units to the left and 2 units up.

By plotting both points on a coordinate plane, you can visualize the line connecting them.

Slope Formula

The slope of a line measures its steepness and direction. It's denoted by the letter \(m\) and is calculated using two points on the line. The formula to find the slope between two points \( (x_1, y_1)\) and \( (x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula gives the ratio of the change in y (vertical change) to the change in x (horizontal change).
For example, using the points (4,2) and (-5,2), substitute into the formula:

• \[ m = \frac{2 - 2}{-5 - 4} = \frac{0}{-9} = 0 \]

A slope of 0 means no vertical change between the points, indicating a horizontal line.

Horizontal Line

A horizontal line is a straight line that runs left to right and has a constant y-coordinate. This means the value of y is the same for all points on the line.
In our example, the line passes through the points (4,2) and (-5,2) with a slope of 0. Since the y-coordinate is always 2, the equation of the line is:
\( y = 2 \)
This implies every point along this line will have a y-coordinate of 2, making it horizontal.
Graphing this line on the coordinate plane helps us visualize its properties.
The line goes through the given points and maintains the same y-coordinate throughout. If you ever find yourself with a slope of 0, you can be sure you're dealing with a horizontal line!