Problem 42 - College Algebra

Question

For the following exercises, find the slope of the line that passes through the given points. \((-5,4)\) and \((2,4)\)

Step-by-Step Solution

Answer

The slope of the line is 0.

Step 1: Understand the Slope Formula

The slope of a line is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

Step 2: Identify the Points

The given points are \((-5, 4)\) which we can call \((x_1, y_1)\), and \((2, 4)\) which we can call \((x_2, y_2)\).

Step 3: Plug the Points into the Slope Formula

Substitute the points into the formula: \( m = \frac{4 - 4}{2 - (-5)} \). This simplifies to \( m = \frac{0}{2 + 5} \).

Step 4: Simplify the Expression

Calculate the denominator \(2 + 5 = 7\), so the equation becomes \( m = \frac{0}{7} \).

Step 5: Interpret the Result

Since the numerator is zero, the slope \( m \) simplifies to 0. This means the line is horizontal.

Key Concepts

Slope Formula

The slope formula is a fundamental concept in mathematics for understanding how steep a line is when graphed in coordinate geometry. The slope of a line is expressed as \( m \), and it's calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula helps find the ratio of the vertical change (Δy) to the horizontal change (Δx) between two points on a line:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of these points.
• The numerator \(y_2 - y_1\) represents the change in the y-values.
• The denominator \(x_2 - x_1\) represents the change in x-values.

A positive slope indicates the line ascends as it moves from left to right, while a negative slope suggests it descends.

Horizontal Line

A horizontal line is a special type of line that runs parallel to the x-axis and maintains a constant y-value throughout. When calculating the slope of a horizontal line:

• The y-coordinates are the same for both points.
• This means \(y_2 - y_1 = 0\), leading to a slope of zero.
• A slope of zero signifies no vertical change as the line stretches out.

Horizontal lines are straightforward to identify in equations. If an equation is in the form of \(y = c\), where \(c\) is a constant, the line is horizontal. For example, the line calculated earlier passing through the points \((-5, 4)\) and \((2, 4)\) is horizontal because the y-value (4) doesn’t change regardless of the x-values.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the branch of mathematics that uses algebraic equations to describe the properties and relationships of geometric figures in a coordinate plane. It allows us to:

• Locate points using coordinates \((x, y)\).
• Calculate the distance between points.
• Find equations of lines.

Understanding coordinate geometry is crucial for solving problems related to slopes. Coordinates help visualize points on a plane, while calculations like the slope provide insight into the nature and behavior of lines. This toolset lets you apply algebraic techniques to geometric problems, such as finding the slope of a line or determining its vertical and horizontal movement. As shown in this exercise, using coordinate geometry simplifies finding whether a line like the one through points \((-5, 4)\) and \((2, 4)\) is horizontal, vertical, or inclined.