Question

In Problems 17-24, plot each pair of points and determine the slope of the line containing the points. Graph the line. $$ (2,3) ;(4,0) $$

Short answer

The slope is \(-\frac{3}{2}\) and the graph of the line passes through (2,3) and (4,0).

Step-by-step solution

- Plot the Points

First, plot the given points on a Cartesian plane. Plot the point (2,3) and then plot the point (4,0).

- Determine the Slope

Use the slope formula to determine the slope of the line connecting the two points. The slope formula is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the values from the given points (2,3) and (4,0):\[ m = \frac{0 - 3}{4 - 2} = \frac{-3}{2} \]Thus, the slope of the line is \(-\frac{3}{2}\).

- Graph the Line

Now graph the line that passes through the points (2,3) and (4,0) with the slope \(-\frac{3}{2}\). Start at one of the points and use the slope to find other points on the line. From (2,3), move down 3 units and to the right 2 units to reach (4,0). Connect these points with a straight line.

Key concepts

Cartesian plane

A Cartesian plane is a two-dimensional plane formed by the intersection of two lines: the x-axis (horizontal) and the y-axis (vertical). These lines are perpendicular to each other and intersect at a point called the origin, denoted as (0,0). The Cartesian plane is divided into four quadrants:
The first quadrant (top right) where both x and y coordinates are positive.
The second quadrant (top left) where x is negative, and y is positive.
The third quadrant (bottom left) where both x and y coordinates are negative.
The fourth quadrant (bottom right) where x is positive, and y is negative.
Each point on the Cartesian plane is represented by an ordered pair (x, y), where 'x' indicates the horizontal position, and 'y' indicates the vertical position.

plotting points

Plotting points on the Cartesian plane is crucial for understanding the position of data. Let's take two points: (2,3) and (4,0) as an example.
To plot (2,3), start at the origin (0,0). Move 2 units to the right along the x-axis and then move up 3 units along the y-axis.
Next, plot the point (4,0). Starting again at the origin (0,0), move 4 units to the right along the x-axis. This point lies on the x-axis since the y-coordinate is 0.
By plotting these points, you can visualize and better understand their positions in relation to each other on the Cartesian plane.

graphing lines

Graphing lines involves connecting points to form a straight line. Using the points (2,3) and (4,0), follow these steps to graph their line:
First, plot both points on the Cartesian plane, as previously discussed.
Next, draw a straight line passing through both points. This line represents all the points that share the same relationship described by the coordinates (2,3) and (4,0).
When graphing lines, you might often use the slope-intercept form of a linear equation, y = mx + b, where 'm' represents the slope, and 'b' represents the y-intercept.
Graphing lines can help understand linear relationships between variables and support visualization of mathematical concepts.

slope formula

The slope of a line is a measure of its steepness and direction. The slope formula, \(\frac{y_2 - y_1}{x_2 - x_1} \), calculates the slope between two points (x₁, y₁) and (x₂, y₂).
Using our example points (2,3) and (4,0), substitute the coordinates into the formula:
\[ m = \frac{0 - 3}{4 - 2} = \frac{-3}{2} \]
This calculation shows that the slope is -3/2. A negative slope indicates that the line is decreasing from left to right.
The slope determines how much the 'y' value changes for a unit change in the 'x' value. Understanding the slope is essential for graphing lines and interpreting their properties in the Cartesian plane.