Question

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

Short answer

The slope is \ -\frac{1}{2}\, and the line passes through \ (-2,3) \ and \ (2,1) \.

Step-by-step solution

- Plot the Points

First, plot the points \((-2,3)\) and \((2,1)\) on the Cartesian plane. Mark the point \((-2,3)\) by starting at the origin, moving 2 units to the left on the x-axis, and then 3 units up on the y-axis. Mark the point \((2,1)\) by starting at the origin, moving 2 units to the right on the x-axis, and then 1 unit up on the y-axis.

- Calculate the Slope

To find the slope \(m\) of the line passing through the points, use the slope formula: \ m = \frac {y_2 - y_1}{x_2 - x_1} \. Here, \(x_1 = -2, y_1 = 3, x_2 = 2, y_2 = 1\). Substitute these values into the slope formula to get \( m = \frac{1 - 3}{2 + 2} = \frac{-2}{4} = -\frac{1}{2} \). So, the slope is \ -\frac{1}{2} \.

- Draw the Line

Using the slope \ -\frac{1}{2} \ and the points you plotted, draw a straight line through the points \((-2,3) \) and \( (2,1) \). The slope \ -\frac{1}{2} \ indicates that for every 2 units you move to the right on the x-axis, you move 1 unit down on the y-axis.

Key concepts

plotting points

Plotting points on a Cartesian plane is the first step to understanding many geometric and algebraic concepts. Start by identifying the coordinates of each point, which are given as pairs \((x, y)\). The first number represents the position on the x-axis (horizontal), and the second number represents the position on the y-axis (vertical).
For example, to plot the point \((-2, 3)\), start at the origin (0,0), move 2 units to the left on the x-axis, and then move 3 units up on the y-axis. To plot the point \((2, 1)\), start again at the origin, move 2 units to the right on the x-axis, and then move 1 unit up on the y-axis.
Plotting both points accurately is crucial for finding the correct slope and drawing the line through them. Once both points are marked, visualize a straight line that would connect them.

slope formula

The slope of a line measures its steepness and direction. It is calculated using the slope formula, which is given as:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \( (x_1, y_1) \) are the coordinates of the first point and \( (x_2, y_2) \) are the coordinates of the second point. The formula calculates the 'rise' (vertical change) over the 'run' (horizontal change) between the two points. For the points \((-2, 3)\) and \((2, 1)\), substitute the values into the formula:
\( m = \frac{1 - 3}{2 - (-2)} = \frac{-2}{4} = -\frac{1}{2} \)
The slope \(-\frac{1}{2}\) means that for every 2 units you move to the right on the x-axis, you will move 1 unit down on the y-axis. Understanding how to use the slope formula helps in determining the steepness and direction of the line.

Cartesian plane

The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). The intersection of these axes is the origin, denoted as \(0,0\).
This plane allows us to graph points, lines, and other geometric figures by using pairs of coordinates. Each point on the Cartesian plane corresponds to a unique pair of coordinates \((x, y)\), where \( x \) specifies the horizontal position, and \( y \) specifies the vertical position. This system makes it easier to visually analyze relationships between different points and lines.
By plotting points and using the slope formula on the Cartesian plane, you can draw and interpret lines, which are fundamental in both algebra and geometry.