Question

Plot the points and draw a line through them. Find the slope of the line passing through the points. $$(0,0),(1,2)$$

Short answer

The slope of the line passing through the points (0,0) and (1,2) is 2.

Step-by-step solution

Plotting the points

The points given are (0,0) and (1,2). Plot these points on a Cartesian coordinate system. The point (0,0) is the origin and the point (1,2) is 1 unit towards right (positive x-direction) and 2 units upwards (positive y-direction) from the origin.

Drawing a line through the points

Draw a line that passes through the two points. This line is the graphical representation of the linear relationship between the x and y values of the points.

Finding the slope of the line

The slope of a line, often represented by the letter 'm', is calculated by the formula \[(y_2 - y_1) / (x_2 - x_1)\]. Here, (x_1,y_1) are the coordinates of one point and (x_2,y_2) are the coordinates of the other point. Using the given points (0,0) and (1,2), the slope is calculated as \[(2 - 0) / (1 - 0) = 2\].

Key concepts

Plotting Points

Understanding how to plot points accurately on a graph is crucial when dealing with many mathematical concepts. To plot a point, you must have a clear understanding of the ordered pair, which consists of an x-coordinate and a y-coordinate. In the ordered pair (x,y), the x-coordinate tells us how far to move horizontally from the origin (0,0), and the y-coordinate tells us how far to move vertically.

Let's consider the ordered pairs given in the exercise, which are (0,0) and (1,2). The first pair represents the origin, where the x and y axes intersect. You simply mark a dot at the center of your coordinate grid. The second point, (1,2), means you move 1 unit to the right (positive x-direction) and 2 units up (positive y-direction) from the origin and place a second dot. Once the points are plotted, they serve as visual cues for drawing lines, curves, or for identifying patterns in data.

Coordinate System

The coordinate system, particularly the Cartesian coordinate system, is a fundamental framework for plotting points, lines, and shapes. It consists of two perpendicular lines, typically labeled as the x-axis (horizontal) and y-axis (vertical). The point where these axes intersect is known as the origin, denoted by (0,0).

The axes divide the plane into four quadrants, labeled counterclockwise as Quadrants I, II, III, and IV. Each quadrant allows for the positive and negative values along the x and y axes. Positive x-values are found to the right of the y-axis, while negative x-values are to the left. Similarly, positive y-values are above the x-axis, and negative y-values are below it. When plotting points, as in the exercise, or when dealing with more complex figures, the coordinate system provides a consistent method to reference locations on the plane.

Linear Equations

Linear equations form straight lines when graphed on a coordinate system. They are beautifully simple yet powerful, characterized by an equation of the form y = mx + b, where 'm' represents the slope and 'b' denotes the y-intercept—the point where the line crosses the y-axis.

In the context of finding slopes, as with the exercise provided, we see a direct application of linear equations. The slope is the measure of the steepness or the incline of the line. It's calculated by taking two distinct points on the line, as you have with (0,0) and (1,2), and using the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\). A positive slope like the one calculated, which is 2, indicates the line rises as it moves from left to right. This concept is pivotal because the slope tells us how much y changes for a unit change in x, allowing us to predict values and understand the rate of change between variables in a variety of applications.