Problem 80 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. Round to two decimal places. $$ (0.00426,-0.00404),(-0.00191,-0.00404) $$

Step-by-Step Solution

Answer

Answer: The slope of the line through the given points is 0.

Step 1: Identify the coordinates

We have the coordinates \((0.00426,-0.00404)\) and \((-0.00191,-0.00404)\), so: - \(x_1 = 0.00426\) - \(y_1 = -0.00404\) - \(x_2 = -0.00191\) - \(y_2 = -0.00404\)

Step 2: Apply the slope formula

Apply the formula for the slope (m): $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Plugging the coordinates into the formula: $$ m = \frac{(-0.00404) - (-0.00404)}{(-0.00191) - (0.00426)} $$

Step 3: Simplify and calculate the slope

Simplify the equation by performing the subtraction: $$ m = \frac{0}{(-0.00617)} $$ Since the numerator is 0, the slope, m, is also 0.

Step 4: Round the answer to two decimal places

The slope, m, is already 0, which doesn't need any rounding. So, the slope of the line through the given points is \(0\).

Key Concepts

Coordinate Geometry

In coordinate geometry, we map out shapes and lines using a system of coordinates, typically expressed as \((x, y)\) pairs.
This system allows us to pinpoint exact locations on a graph by specifying horizontal and vertical positions. The horizontal position is indicated by the value of \(x\), while the vertical position is expressed as \(y\).
By using this method, we can easily visualize math problems, and solutions related to lines, angles, and shapes.
Coordinate geometry is fundamental in understanding how to calculate the distance between points, find midpoints, and determine slopes of lines.
In this specific exercise, we are focusing on the slope of a line, determined using two specified points in this coordinate system. These two points are essential as they provide the necessary data to calculate how steep or flat a line on the graph is.

Slope Formula

The slope of a line is a critical concept that represents its steepness, indicating the rate at which \(y\) changes with respect to changes in \(x\).
The slope is often symbolized as \(m\) and calculated with the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1} \]where

• \(y_1\) and \(y_2\) are the \(y\)-coordinates of two points on the line, and
• \(x_1\) and \(x_2\) are the \(x\)-coordinates of the same points.

Using this formula, we can clearly determine the line's direction:

• If \(m > 0\), the line is ascending.
• If \(m < 0\), the line is descending.
• If \(m = 0\), the line is horizontal.
• A vertical line has an undefined slope, as it would require division by zero.

For the given coordinates \((0.00426,-0.00404)\) and \((-0.00191,-0.00404)\), the calculation yields \(m = 0\), indicating a horizontal line where \(y\) does not change regardless of \(x\).

Mathematical Calculations

Mathematical calculations are essential in breaking down complex problems into understandable parts, especially with a concept like the slope.
To find the slope, we start by identifying two points on the coordinate plane. For this problem, these points are \((0.00426, -0.00404)\) and \((-0.00191, -0.00404)\).
Next, we plug these coordinates into the slope formula:\[m = \frac{-0.00404 - (-0.00404)}{-0.00191 - 0.00426} \]By performing the necessary mathematical operations:

• The numerator \((-0.00404) - (-0.00404)\) simplifies to 0, showing there's no change in \(y\).
• The denominator, \((-0.00191) - 0.00426\), simplifies to -0.00617.

While the denominator is not zero, the overall fraction results in 0 because the numerator is zero, indicating no slope.
These steps demonstrate how precise calculations can lead to a clear understanding of a mathematical problem.