Question

Plot each pair of points and determine the slope of the line containing the points. Graph the line. $$ (-3,-1) ;(2,-1) $$

Short answer

Slope = 0; the line is horizontal through y = -1.

Step-by-step solution

Plot the Points

First, locate and plot the points (-3, -1) and (2, -1) on a coordinate plane. Point A is at (-3, -1) and Point B is at (2, -1).

Connect the Points

Draw a line connecting the two plotted points A (-3, -1) and B (2, -1).

Determine the Slope

The slope of a line through two points (x1, y1) and (x2, y2) is given by the formula: \[ m = \frac{y2 - y1}{x2 - x1} \] Substitute the given points: \[ m = \frac{-1 - (-1)}{2 - (-3)} = \frac{0}{5} = 0 \] So, the slope is 0.

Graph the Line

Since the slope is 0, the line is horizontal. A horizontal line passing through y = -1 should be drawn.

Key concepts

coordinate plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves to represent mathematical concepts visually. This plane is defined by two perpendicular number lines intersecting at a point called the origin, which has coordinates (0, 0). These two number lines are called the x-axis (horizontal) and the y-axis (vertical). Each point on the coordinate plane is identified by a pair of numbers, written as (x, y), where x represents the horizontal position and y represents the vertical position. For example, the point (-3, -1) has an x-coordinate of -3 and a y-coordinate of -1, meaning it’s located 3 units to the left of the origin and 1 unit down.

plotting points

Plotting points on the coordinate plane is a fundamental skill in graphing. To plot a point, follow these steps:

• First, identify the x-coordinate of the point. This tells you how far left or right to move from the origin.
• Next, identify the y-coordinate. This tells you how far up or down to move from the x-coordinate.
• Finally, mark the point where these coordinates intersect.

For instance, to plot the points (-3, -1) and (2, -1), we:
1. Locate -3 on the x-axis and move 3 units left.
2. Locate -1 on the y-axis and move 1 unit down.
3. Mark the point where these positions intersect.
Repeating these steps for the point (2, -1) tells you to move right 2 units and down 1 unit.

horizontal line

A horizontal line is one of the simplest types of lines on the coordinate plane. It runs parallel to the x-axis, meaning it has the same y-coordinate for all points on the line. In our exercise, points (-3, -1) and (2, -1) both have a y-coordinate of -1. Drawing a line through these points shows that the line does not rise or fall, confirming it as horizontal. Horizontal lines are unique because their slope is always 0, indicating zero vertical change as one moves along the line.

slope formula

The slope of a line measures its steepness and direction. It's calculated using the slope formula:

\[ m = \frac{y2 - y1}{x2 - x1} \]

where (x1, y1) and (x2, y2) are two distinct points on the line. The slope \( m \) is the ratio of the vertical change (rise) to the horizontal change (run) between the points.

• First, find the differences between the y-coordinates (rise).
• Next, find the differences between the x-coordinates (run).
• Finally, divide the rise by the run.

In our exercise:
1. Use points (-3, -1) and (2, -1).
2. Calculate the rise: \( -1 - (-1) = 0 \).
3. Calculate the run: \( 2 - (-3) = 5 \).
4. Determine the slope: \( m = \frac{0}{5} = 0 \).
This confirms the line is horizontal with a slope of 0.