Question

Plot the points and find the slope of the line passing through them. $$ (3,-2),(4,-2) $$

Short answer

The slope of the line passing through the points (3, -2) and (4, -2) is 0.

Step-by-step solution

Plot the points

Plot the points (3, -2) and (4, -2) on a two-dimensional graph. The x-coordinates of the points are 3 and 4, and both have a y-coordinate of -2. This means both points lie on the same horizontal line.

Finding the slope

Slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \( \frac{{y_2 - y_1}}{{x_2 - x_1}} \). Substituting the coordinates of the given points into the slope formula, we get \( \frac{{-2 - (-2)}}{{4 - 3}} = \frac{{0}}{{1}} \).

Result

The slope of the line passing through points (3, -2) and (4, -2) is 0. This indicates a horizontal line, which confirms our graph plotted in step 1.

Key concepts

Plotting Points

When beginning to understand the geometry of a line, one of the most essential skills is plotting points. To plot a point like \( (3, -2) \), we find the coordinate on the x-axis (horizontal line) and the coordinate on the y-axis (vertical line). Firstly, move 3 units to the right from the origin (the center where x and y axes intersect) since the x-coordinate is 3. Then, move 2 units down since the y-coordinate is -2. Place a dot at this intersection point. Repeat the same process for the second point, \( (4, -2) \), which will be just one unit to the right of the first point since the x-coordinate is 4, but still on the same horizontal level, since the y-coordinate is -2. After plotting both points, you will notice that they lie in a straight line that is parallel to the x-axis.

The ability to visualize the placement of points on the Cartesian plane is crucial, as it aids in understanding the slope, orientation, and relation between points on a graph.

Slope Formula

The slope of a line is a numerical measure of its steepness, usually denoted as \( m \). To calculate the slope between two points, we employ the slope formula \(\frac{{y_2 - y_1}}{{x_2 - x_1}}\). This formula essentially describes the rate at which the y-value changes with respect to the x-value. When the two points \( (x_1, y_1) \) and \( (x_2, y_2) \) are known, you can find the slope by subtracting the y-values and dividing by the difference in x-values. For example, given the points \( (3, -2) \) and \( (4, -2) \), we apply the formula to get \( \frac{{-2 - (-2)}}{{4 - 3}} = \frac{{0}}{{1}} \), rendering the slope as 0. This computation shows that for every one unit increase in \( x \), the change in \( y \), and thus the elevation of the line, does not change -- indicating a flat, or horizontal, orientation.

Horizontal Line

A horizontal line is one that runs left to right across the plane and has a slope of 0. This is because there is no vertical change as one moves along the line; the y-coordinates are constant, which translates into a 0 rise over any run you might calculate between two points on the line. In our exercise, once the points are plotted and connected, you'll notice that they create a perfectly horizontal line. The slope formula confirms this, since the calculation showed a slope of 0. Such lines are parallel to the x-axis and are a visual representation of a constant function, where \( y \) does not change no matter the value of \( x \). Recognizing a horizontal line is beneficial, as it is a basic concept in geometry that can help with understanding more complex mathematical ideas.