Problem 73 - Algebra 1: Concepts and Skills

Question

Find the slope of the line that passes through the points. \((4,1)\) and \((6,1)\)

Step-by-Step Solution

Answer

The slope of the line that passes through the points \((4,1)\) and \((6,1)\) is 0.

Step 1: Identify the given points

The two given points are \((4,1)\) and \((6,1)\). Identify \(x1 = 4\), \(y1 = 1\), \(x2 = 6\), and \(y2 = 1\).

Step 2: Substitute into the slope formula

Substitute the coordinates of the given points into the slope formula: \(m = \frac{y2 - y1}{x2 - x1}\) becomes \(m = \frac{1 - 1}{6 - 4}\).

Step 3: Simplify the equation

Simplify the equation \(m = \frac{0}{2}\) which gives \(m = 0\).

Key Concepts

Understanding the Slope Formula

The concept of a slope is central to understanding how lines behave on a graph. The slope essentially tells us how steep or flat a line is.
It's calculated using the **slope formula**, which is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here,

• \(m\) represents the slope of the line,
• \( (x_1, y_1) \) are the coordinates of the first point, and
• \( (x_2, y_2) \) are the coordinates of the second point.

The formula calculates the change in the vertical direction, or "rise," over the change in the horizontal direction, or "run."
A positive slope means the line goes up from left to right; a negative slope means it goes down; a **zero slope** indicates a flat, horizontal line, like in our example where the slope is 0.

Exploring Coordinate Points

Coordinate points are fundamental in plotting and understanding lines on a graph. Each point is represented as (x, y), where

• \(x\) gives the position on the horizontal axis,
• \(y\) gives the position on the vertical axis.

In the example, the points \((4, 1)\) and \((6, 1)\) describe two positions on the graph.
Both points have the same \(y\) value, indicating they lie on the same horizontal level, which reflects why the slope is zero. When the \(y\) coordinates remain consistent, it signals a horizontal line.

The Role of Linear Equations

A linear equation describes a straight line on a graph. The formula for a linear equation in its simplest form is \(y = mx + b\), where

• \(m\) is the slope,
• \(b\) is the y-intercept, which is where the line crosses the y-axis.

In linear equations, every change in \(x\) results in a direct and predictable change in \(y\). This consistency is defined by the slope \(m\).
For the equation of a horizontal line, the equation simplifies because the slope \(m\) is 0. Therefore, the line's equation becomes \(y = b\). In this example, both points have a \(y\)-value of 1, meaning the equation of the line is \(y = 1\), which takes on this simple horizontal form.