Problem 65 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. $$ (2,3),(10,3) $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 0 (horizontal line).

Step 1: Identify the coordinates of the two points

Assigning the points on the Cartesian plane, we would have: Point 1: (2, 3) which has coordinates \(x_1 = 2\) and \(y_1 = 3\) Point 2: (10, 3) which has coordinates \(x_2 = 10\) and \(y_2 = 3\)

Step 2: Apply the slope formula

Now we'll use the slope formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Step 3: Substitute the coordinates of the points into the formula

Let's plug in the coordinates of the points we found earlier into the formula: $$ m = \frac{3 - 3}{10 - 2} $$

Step 4: Solve for the slope m

Now, we simply need to calculate the result: $$ m = \frac{0}{8} = 0 $$

Step 5: Interpret the result

Since the slope of the line is 0, we can conclude that it's a horizontal line.

Key Concepts

Understanding Coordinates

Coordinates are fundamental for locating points on a Cartesian plane. Each point on this grid is defined by a pair of numbers: the \(x\)-coordinate and the \(y\)-coordinate.
These values tell you how far along the horizontal (\(x\)-axis) and vertical (\(y\)-axis) a point is.
To better visualize:

• The first number in a pair, like 2 in (2,3), is the \(x\)-coordinate.
• The second number, such as 3 in both (2,3) and (10,3), is the \(y\)-coordinate.

So, the coordinates for the points are (2,3) - meaning 2 steps right and 3 steps up from the origin - and (10,3) - meaning 10 steps right and 3 steps up.
This helps us place points precisely and is essential for applying the slope formula.

Utilizing the Slope Formula

The slope formula is a crucial tool for determining how steep a line is. It compares the vertical change (rise) to the horizontal change (run) between two points.
The formula is represented as: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where:

• \(y_1\) and \(y_2\) are the y-values of the two points.
• \(x_1\) and \(x_2\) are the x-values of the two points.

The slope \(m\) shows the line's inclination. A positive \(m\) means the line goes upward, while a negative \(m\) means it descends. If \(m = 0\), the line is perfectly horizontal, indicating no incline.
By applying this formula, we get a clear mathematical picture of the line between our points.

Exploring Horizontal Lines

A horizontal line is unique in that it stretches from left to right without any vertical climb or descent. Such lines have several distinguishing features:

• The \(y\)-coordinate remains constant, meaning all points on the line share the same \(y\)-value.
• The slope of a horizontal line is always 0, signifying no vertical change as you move across the line.

For this exercise, the points (2,3) and (10,3) share the same \(y\)-coordinate.
This constant \(y\)-value indicates the line is horizontal.
No matter how far apart the points are on the \(x\)-axis, the height doesn’t alter.
Understanding horizontal lines helps in recognizing and interpreting graphs, allowing us to see when a relationship has no change or growth in \(y\).