Question

Find the slope of the lines passing through each pair of points. (Lesson 4-5) $$(8,4),(-2,4)$$

Short answer

The slope of the line is 0.

Step-by-step solution

Identify the Points

We begin by identifying the given points. The points are given as \((x_1, y_1) = (8, 4)\) and \((x_2, y_2) = (-2, 4)\).

Recall the Slope Formula

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

Substitute the Values

Substitute the values from the points into the slope formula: \[ m = \frac{4 - 4}{-2 - 8} \]

Calculate the Difference in the Numerator

Calculate the difference in the numerator of the slope equation: \(4 - 4 = 0\).

Calculate the Difference in the Denominator

Calculate the difference in the denominator of the slope equation: \(-2 - 8 = -10\).

Compute the Slope

With the differences calculated, find the slope: \[ m = \frac{0}{-10} = 0 \].

Key concepts

Coordinate Points

In the world of geometry, coordinate points are essential to understanding the layout of shapes and lines on a plane. They are a pair of numbers used to determine the location of a point in a two-dimensional space. Each coordinate point is written as

• the first value represents the position along the horizontal axis, known as the x-coordinate.
• the second value represents the position along the vertical axis, known as the y-coordinate.

For example, in the points

• a given coordinate \((x_1, y_1)\) (our first point, (8, 4) here) tells us the point is 8 units along the x-axis and 4 units along the y-axis.
• the other coordinate \((x_2, y_2)\) (the second point, (-2, 4)) means it's -2 units horizontally and 4 units vertically.

Coordinate points allow us to easily plot and find relationships between points on a graph, such as determining the slope between two points.

Slope Formula

The slope formula is key to determining the steepness of a line connecting two points. Slope is often represented by the letter \(m\), and can be thought of as "rise over run." It gives us a measure of how much a line goes up or down based on its horizontal span. The formula for slope, when given two points \((x_1, y_1)\) and \((x_2, y_2)\), is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula tells us:

• The difference in the y-values, \(y_2 - y_1\), gives us the "rise," or vertical change.
• The difference in x-values, \(x_2 - x_1\), gives us the "run," or horizontal change.

By dividing the change in y by the change in x, we find how much the y-value increases (or decreases) per unit of x. Applying this formula to our example points ((8,4) and (-2,4)), we calculated a slope of 0, which tells us something specific about this line.

Horizontal Line

A horizontal line is unique in the realm of geometry as it has a slope of 0. This means there is no vertical change as you move along the line, it always stays constant regardless of how far you go left or right. For our example, after using

• the slope formula with points ((8, 4) and (-2, 4)), yielding a slope of 0, signifies that y-values at both points are equal.
• They start and end on the same y-level (y = 4), showing no rise, just run.

In practical terms:

• A horizontal line acts parallel to the x-axis and can represent steady conditions like a flattened road or a calm sea level.
• This unmistakable trait makes them easy to identify on a graph.

Recognizing a horizontal line from its slope helps students understand not only geometry, but also its applications across various fields such as physics and engineering.