Problem 42 - A Graphical Approach to College Algebra

Question

Find the slope (if defined) of the line that passes through the given points. \((-4,-3)\) and \((5,0)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Points

We are given two points through which the line passes: Point 1: \((-4, -3)\)Point 2: \((5, 0)\).

Step 2: Apply the Slope Formula

The formula for the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Step 3: Substitute the Values

Substitute the coordinates of the given points into the formula:\[m = \frac{0 - (-3)}{5 - (-4)} = \frac{0 + 3}{5 + 4}\]

Step 4: Simplify the Expression

Simplify the fraction:\[m = \frac{3}{9} = \frac{1}{3}\]Therefore, the slope \( m \) of the line is \( \frac{1}{3} \).

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that represent straight lines when plotted on a graph. They are usually written in the form \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) is the slope of the line, which shows the line's steepness.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

These elements together ensure that as the value of \( x \) changes, the value of \( y \) adjusts accordingly, maintaining a constant rate of proportionality dictated by the slope \( m \). Linear equations are fundamental in understanding relationships in coordinate geometry and are utilized in various fields including physics, economics, and engineering.
Once you determine the slope, you can easily write the equation of the line, if the intercept is known or provided.
When solving problems involving linear equations, finding the slope is a key first step.

Exploring the Slope Formula

The slope formula is essential in determining the steepness of a line between two points on a graph. It gives clear insight into how much \( y \) changes for a given change in \( x \). The formula for calculating the slope (\( m \)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This expression calculates the "rise over run," reflecting:

• The change in \( y \) (vertical change)
• Divided by the change in \( x \) (horizontal change)

In the given problem, using the points \((-4, -3)\) and \((5, 0)\), the slope is calculated as follows: \( m = \frac{0 - (-3)}{5 - (-4)} = \frac{3}{9} = \frac{1}{3} \).
This slope value \( \frac{1}{3} \) indicates that for each unit increase in \( x \), \( y \) increases by \( \frac{1}{3} \) of a unit.
The slope is a critical part of understanding the characteristics of linear equations and how they relate two variables through a linear relationship.

Introduction to Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to study geometric shapes, sizes, and positions by using an algebraic approach. Points are plotted on a two-dimensional plane, referred to as the Cartesian coordinate system.
The coordinates of a point are defined as \((x, y)\), where:

• \( x \) is the horizontal position (abscissa)
• \( y \) is the vertical position (ordinate)

Through coordinate geometry, we can analyze the behavior of geometric figures through algebraic representations.
In the exercise, identifying the correct coordinates of the points \((-4, -3)\) and \((5, 0)\) is the first step in solving problems involving lines and finding the slope.
This approach not only assists in precisely finding geometric relationships but also in visualizing how equations translate into shapes and lines on graphs. Coordinate geometry thus bridges the gap between algebra and geometry, making it easier to understand spatial relationships.