Problem 2 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (2,2),(4,2) $$

Step-by-Step Solution

Answer

The slope is 0.

Step 1: Understand the Slope Formula

The formula for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]This formula calculates how much the line rises vertically for a given horizontal change between the two points.

Step 2: Define the Coordinates

Identify the coordinates of the given points. We have the points \((x_1, y_1) = (2, 2)\)and \((x_2, y_2) = (4, 2)\).Plug these coordinates into the slope formula.

Step 3: Substitute the Values

Substitute the values into the slope formula:\[m = \frac{2 - 2}{4 - 2} = \frac{0}{2}.\]

Step 4: Calculate the Slope

Calculate the expression from Step 3. Since the numerator is zero:\[m = \frac{0}{2} = 0.\]This means the line is horizontal, indicating zero vertical change between the points.

Key Concepts

Slope Formula

The slope formula is a simple yet powerful tool in geometry to determine the steepness or inclination of a line on a graph. It tells us how much a line moves up or down for a certain distance moved horizontally. To find the slope of a line passing through two points, we use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\] where

• \(m\) represents the slope.
• \((x_1, y_1)\) and \((x_2, y_2)\) are coordinate points on the line.
• The numerator \((y_2 - y_1)\) measures the vertical change (rise) between the two points.
• The denominator \((x_2 - x_1)\) measures the horizontal change (run).

If the slope \(m\) is positive, the line ascends as it moves from left to right. A negative slope means the line descends. If the slope is zero, the line is horizontal, indicating no vertical change.

Horizontal Line

A horizontal line is a straight line that goes from left to right across a graph. When we find that the slope \(m\) is zero, it means the line does not rise or fall, indicating no vertical change between any two points on the line. This is characteristic of horizontal lines.Horizontal lines have a constant \(y\)-coordinate. In the coordinates \((2, 2)\) and \((4, 2)\), you can see that both points share the same \(y\)-coordinate, which is 2. This consistency results in no vertical change, thus a zero slope:- The expression \(y_2 - y_1 = 2 - 2 = 0\)Since there is no rise, horizontal lines appear flat on a graph, and this flatness is reflected in the slope formula by the zero result.

Coordinate Points

Coordinate points are essential in geometry as they specify the exact position of a point on the Cartesian plane. Each point is defined by a pair of numbers written in parentheses, like \(x, y\). The first number indicates the horizontal position (x-coordinate), and the second indicates the vertical position (y-coordinate).When you are trying to find the slope of a line between two points, you first identify the coordinates of both points, for example:

• Point 1: \( (x_1, y_1) = (2,2) \)
• Point 2: \( (x_2, y_2) = (4,2) \)

These values are substituted into the slope formula to determine how steep or flat the line is. Understanding coordinate points allows you to visualize and calculate various properties of lines, such as slopes and intercepts, on a graph.