Problem 11 - Brief Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. \((0,-1)\) and \((4,-1)\)

Step-by-Step Solution

Answer

The slope of the line is 0, indicating it is horizontal.

Step 1: Identify the formula for slope

The slope of a line through two points, \((x_1, y_1)\) and \((x_2, y_2)\), is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).Identify the points: \((x_1, y_1) = (0, -1)\) and \((x_2, y_2) = (4, -1)\).

Step 2: Substitute the coordinates into the formula

Insert the coordinates from the points into the slope formula: \(m = \frac{-1 - (-1)}{4 - 0}\).

Step 3: Calculate the slope

Simplify the expression: \(m = \frac{-1 + 1}{4 - 0} = \frac{0}{4} = 0\).

Step 4: Interpret the result

The slope \(m = 0\) indicates that the line is horizontal, meaning it has no vertical change as it moves along the horizontal axis.

Key Concepts

Understanding Linear Equations

Linear equations are equations that graph to a straight line when plotted on a coordinate plane. In their simplest form, they express the relationship between two variables, typically x and y. The general form of a linear equation in two variables is \(y = mx + b\), where:

• \(m\) represents the slope, or the steepness of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

In the context of slope calculation from given points, the slope \(m\) can tell us how much y increases (or decreases) for a one-unit increase in x. For example, if \(m = 2\), it means that for every increase of 1 unit in x, y will increase by 2 units.

Linear equations are crucial in algebra and are used to model a wide range of real-world situations, from business forecasting to physics applications. A key characteristic of linear equations is their constant rate of change, signified by the slope.

Exploring Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebraic methods to solve geometric problems. It uses a coordinate plane to illustrate points, lines, and shapes. The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

• The location of any point on this plane is described by an ordered pair \((x, y)\).
• The distance between points and their slopes are key focus areas within coordinate geometry.

In coordinate geometry, the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) helps to determine how steep a line is between two points \((x_1, y_1)\) and \((x_2, y_2)\). This approach simplifies the process of examining the relationships between various parts of geometric shapes.

Coordinate geometry serves as a bridge between algebra and geometry, providing hefty tools to not only prove geometric theorems but also to solve complex algebraic equations.

Insights into Horizontal Lines

Horizontal lines are lines that run parallel to the x-axis in a coordinate plane. They are unique because they maintain a constant y-value across all points. This means there is no change in the y-coordinate regardless of the x-coordinate value.

• The equation of a horizontal line takes the form \(y = c\), where \(c\) is a constant.
• For a horizontal line, the slope \(m\) is always 0.

This is due to the fact that the formula for slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) results in zero when \(y_2 = y_1\) because the difference is zero. This indicates there is no vertical change as you move along the horizontal direction.

Horizontal lines are a fundamental concept when considering linear relationships without vertical movement. They are especially useful in various disciplines, including economics, where a fixed price is represented by a horizontal line on a price-quantity graph.