Problem 13 - College Algebra with Modeling and Visualization

Question

If possible, find the slope of the line passing through each pair of points. $$ (0.2,-0.1),(-0.3,0.4) $$

Step-by-Step Solution

Answer

The slope of the line is \(-1\).

Step 1: Identify the Coordinates of the Points

The points given are \((0.2, -0.1)\) and \((-0.3, 0.4)\). Let's label them as \((x_1, y_1) = (0.2, -0.1)\) and \((x_2, y_2) = (-0.3, 0.4)\).

Step 2: Recall the Slope Formula

The formula for the slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Step 3: Substitute the Coordinates into the Slope Formula

Substitute \(y_1 = -0.1\), \(y_2 = 0.4\), \(x_1 = 0.2\), and \(x_2 = -0.3\) into the formula:\[m = \frac{0.4 - (-0.1)}{-0.3 - 0.2}\]

Step 4: Calculate the Numerator of the Slope Formula

Calculate the difference \(y_2 - y_1\):\[0.4 - (-0.1) = 0.4 + 0.1 = 0.5\]

Step 5: Calculate the Denominator of the Slope Formula

Calculate the difference \(x_2 - x_1\):\[-0.3 - 0.2 = -0.5\]

Step 6: Find the Slope by Dividing the Numerator by the Denominator

Divide the result of the numerator by the result of the denominator to find the slope:\[m = \frac{0.5}{-0.5} = -1\]

Step 7: Interpret the Slope

The slope of the line passing through the points \((0.2, -0.1)\) and \((-0.3, 0.4)\) is \(-1\). This means that for every unit increase in \(x\), \(y\) decreases by 1 unit.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that deals with some of the most fundamental concepts of geometry and algebra, by placing them within a coordinate system. This allows us to visualize mathematical problems in a spatial context.
Coordinate systems involve an x-axis and a y-axis that intersect perpendicularly at the origin (0,0), creating a plane upon which any point can be represented as a pair of numbers. These numbers, known as coordinates, detail the precise location of a point relative to the origin.
Consider the points \(0.2, -0.1\) and \(-0.3, 0.4\). The first number in each pair is the x-coordinate, and the second is the y-coordinate. These coordinates help us track movement on the plane—for example, when finding a line that connects two points or determining the slope based on those coordinates.

Slope Formula

The Slope Formula is essential for determining the steepness or incline of a line connecting two points in a coordinate plane. It tells us how much a line rises or falls as it moves from left to right across the graph.
Formulaically, if you have two points \(x_1, y_1\) and \(x_2, y_2\), the slope \(m\) of the line passing through those points is calculated as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
This formula effectively measures the change in the y-coordinates (vertical change) over the change in the x-coordinates (horizontal change). A positive slope means the line is upward sloping, while a negative slope indicates it slopes downward. A zero slope signifies a horizontal line, which indicates no vertical change, and an undefined slope corresponds to a vertical line.

Calculating Slope

Calculating the slope involves substituting the coordinates of two points into the Slope Formula. It is a straightforward process but essential for understanding the relationship between points on a line.
For example, using our points \(0.2, -0.1\) and \(-0.3, 0.4\), first calculate the difference in the y-values, \((0.4 - (-0.1))\), which equals 0.5. Then, find the difference in the x-values, \((-0.3 - 0.2)\), which results in -0.5.
Finally, divide the difference in the y-values by the difference in the x-values to find the slope, using the formula:\[m = \frac{0.5}{-0.5} = -1\]
Thus, we find the slope is \(-1\), indicating a decrease in y by 1 unit for every unit increase in x, clearly showing a downward slope.