Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Plot the points and find the slope of the line passing through the points. \((0,-10),(-4,0)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((0,-10)\) and \((-4,0)\) is -2.5.

Step 1: Plot the Points

First, plot the points \( (0,-10) \) and \( (-4,0) \) on a Cartesian coordinate system. Mark each point, respectively.

Step 2: Draw a Line Through the Points

Once the points are plotted, the next step is to draw a straight line that passes through both points. This line represents the line whose slope to be determined.

Step 3: Calculate the Slope

The slope of the line (m) is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the values in the formula: \( m = \frac{0 - (-10)}{-4 - 0} = \frac{10}{-4} = -2.5\).

Key Concepts

Plotting Points

Plotting points is an essential skill in graphing that helps visualize relationships between numbers. To plot a point like \((0, -10)\) or \((-4, 0)\), we use a simple method. The first number, called the x-coordinate, tells us how far to move horizontally from the origin (which is the point \((0, 0)\)). The second number, the y-coordinate, indicates how far to move vertically. For instance, starting at the origin, move 0 units right or left for \((0, -10)\), then move 10 units downward. This places the point exactly at \((0, -10)\).
Similarly, for \((-4, 0)\), move 4 units to the left from the origin, but don’t move up or down because the y-coordinate is 0. This places the point right on the x-axis. Plotting these points accurately sets the stage for creating lines and understanding slopes.

Cartesian Coordinate System

The Cartesian Coordinate System is a standard method for plotting points and equations on a plane. Named after the mathematician René Descartes, this system uses two perpendicular axes: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin, which is the point \((0,0)\).
This system divides the plane into four quadrants, which help locate points. Here’s a quick breakdown:

• Points in Quadrant I have positive x and y coordinates.
• Quadrant II has negative x and positive y coordinates.
• Quadrant III has negative x and y coordinates.
• Quadrant IV has positive x and negative y coordinates.

Understanding this coordinate system is key when plotting points like \((0, -10)\) and \((-4, 0)\), and helps visualize the concepts of lines and slopes.

Line Equation

A line equation provides a simple way to describe a straight line on a graph. One common form of this equation is the slope-intercept form: \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, where the line crosses the y-axis.
When we have two points, such as \((0, -10)\) and \((-4, 0)\), we can find the line's slope using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Our earlier calculation showed the slope is \(-2.5\). This means for every unit increase in x, y decreases by 2.5 units.
Knowing how to calculate and use the slope helps us build the line’s equation. Since one point is \((0, -10)\), our y-intercept \(b\) is \(-10\). Thus, the equation of the line can be written as \(y = -2.5x - 10\). Understanding line equations is essential for predicting and describing mathematical relationships in a visual form.