Problem 41 - Algebra 1

Question

Without calculating, state whether the slope of the line through the points is positive, negative, zero, or undefined. (6,1),(3,1)

Step-by-Step Solution

Answer

The slope of the line through points (6,1) and (3,1) is zero.

Step 1: Identify the coordinates

The first coordinate is (6,1) and the second is (3,1). The coordinates are given in the form (x,y). Thus, for the first point, x1=6 and y1=1. For the second point, x2=3 and y2=1.

Step 2: Apply the formula for slope

The slope m of a line through points (x1,y1) and (x2,y2) is calculated as m = (y2-y1)/(x2-x1). Substituting the given values into the formula, we get m = (1-1)/(3-6).

Step 3: Simplify the expression

Simplify the numerator and the denominator separately. Here, the numerator (y2-y1) equals 0 because 1 minus 1 is 0. The denominator (x2-x1) equals -3 because 3 minus 6 is -3.

Step 4: Conclude the type of the slope

As the numerator (vertical change) is zero and the denominator (horizontal change) is not zero, the slope of the line is zero which implies the line is horizontal.

Key Concepts

Understanding Coordinates

Coordinates are a fundamental concept in mathematics and are essential for identifying locations on a plane. In a two-dimensional plane, coordinates are usually given as ordered pairs (x, y), where 'x' is the horizontal position and 'y' is the vertical position.
Every point on the plane can be uniquely identified using these coordinates. For instance, if you have a point (6, 1), it means that:

• Move 6 units along the x-axis from the origin,
• Then, move 1 unit up the y-axis.

This method applies similarly to any point, making coordinates a universal way of describing positions in mathematics.

Recognizing a Horizontal Line

A horizontal line is characterized by a constant y-value across all its points. Geometrically, it runs parallel to the x-axis, lying level without any tilt.
When a line passes through two points with identical y-coordinates, such as (6, 1) and (3, 1), you have a horizontal line. This means:

• The vertical position (y-value) doesn’t change as you move along the line.
• Only the x-value, or horizontal position, changes.

This is why horizontal lines have a slope of zero, as there is no rise, only a run, echoing the fundamental trait of uniformity in height across its length.

Applying the Slope Formula

The slope of a line is a measure of its steepness or inclination and is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \(m\) represents the slope, - \(y_2\) and \(y_1\) are the y-coordinates of two points on the line, - \(x_2\) and \(x_1\) are the x-coordinates.
For the given points (6,1) and (3,1), notice how the y-coordinates are the same. Substituting into the formula gives: \[ m = \frac{1 - 1}{3 - 6} = \frac{0}{-3} \] Because the numerator (change in y) is zero, the slope is zero. This situation signifies a perfectly horizontal line, where there's no vertical change.