Problem 9 - Applied Calculus

Question

Find the slope (if it is defined) of the line determined by each pair of points. $$ (-4,0) \text { and }(2,2) $$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{1}{3}\).

Step 1: Understand the Slope Formula

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

Step 2: Plug in the Coordinates of the Points

Substitute the given points \((-4, 0)\) and \((2, 2)\) into the slope formula. Here, \(x_1 = -4, y_1 = 0, x_2 = 2,\) and \(y_2 = 2\).

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness and is represented by the letter \( m \). To find the slope between two points, you use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This equation takes the vertical change (difference in the \( y \)-coordinates) and divides it by the horizontal change (difference in the \( x \)-coordinates). This ratio tells us how much the line rises for each unit it runs.When you plug in the coordinates of the points

• Point 1: \((-4, 0)\)
• Point 2: \((2, 2)\)

we substitute them into the formula:\[ m = \frac{2 - 0}{2 - (-4)} \]This calculation gives:\[ m = \frac{2}{6} = \frac{1}{3} \]Thus, the slope of the line that connects these two points is \( \frac{1}{3} \). The positive value indicates that the line ascends from left to right.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra with geometry through graphs and coordinates. It allows us to represent geometric figures like lines and curves using coordinate planes.
This branch of mathematics utilizes a system where every point is defined by ordered pairs \((x, y)\), known as coordinates.This method is highly versatile. It helps in solving geometric problems algebraically and visualizing algebraic equations geometrically.

• The horizontal line on the plane is the \( x \)-axis.
• The vertical line is the \( y \)-axis.
• The intersection point where both meet is the origin, \((0,0)\).

When given a pair of points, like in our exercise, coordinate geometry makes it possible to find the slope, distances, and midpoint values easily. Understanding this concept is crucial as it forms the foundation for more complex topics in mathematics.

Linear Equations

Linear equations are algebraic expressions that form straight lines when plotted on a graph. They are typically written in the standard format \( y = mx + c \), where:

• \( m \) represents the slope of the line
• \( x \) is the independent variable
• \( y \) is the dependent variable
• \( c \) is the y-intercept, which is the point where the line crosses the \( y \)-axis

Linear equations are fundamental because they describe a constant rate of change between two variables, making them useful in numerous real-life applications. For example, understanding distances, speed relations, or predicting trends.In our exercise, after finding the slope \( \frac{1}{3} \) using the two points given, one could further develop the full linear equation of the line passing through those points by using one point and the slope-intercept form \( y = mx + c \). With this capability, the power of linear equations becomes apparent as it permits prediction and analysis of linear relationships efficiently.