Problem 40 - College Algebra

Question

For the following exercises, find the slope of the line that passes through the given points. \((5,4)\) and \((7,9)\)

Step-by-Step Solution

Answer

The slope is \(\frac{5}{2}\).

Step 1: Identify the Points

The two given points are \((x_1, y_1) = (5, 4)\) and \((x_2, y_2) = (7, 9)\).

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 \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given values into the formula: \(m = \frac{9 - 4}{7 - 5}\).

Step 3: Calculate the Differences

Subtract to find the differences: \(y_2 - y_1 = 9 - 4 = 5\) and \(x_2 - x_1 = 7 - 5 = 2\).

Step 4: Find the Slope

Divide the differences to find the slope: \(m = \frac{5}{2}\). The slope of the line through the points \((5, 4)\) and \((7, 9)\) is \(\frac{5}{2}\).

Key Concepts

Slope Formula

Understanding the slope formula is key to mastering the concept of the slope of a line. The slope tells us how steep a line is and the direction it slants. It is often denoted by the letter \(m\). The formula used to find the slope of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula calculates the change in the \(y\)-coordinates (also known as the rise) over the change in the \(x\)-coordinates (the run).

When applying the slope formula:

• First, subtract the \(y\)-coordinate of the first point from the \(y\)-coordinate of the second point.
• Next, subtract the \(x\)-coordinate of the first point from the \(x\)-coordinate of the second point.
• Finally, divide these results to find the slope.

In simpler terms, the slope is simply the rate of change between two points on a line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It allows for the precise description of geometric shapes and positions on a graph through the use of coordinates, which are numerical values on a plane.

This geometry system uses two axes: the horizontal \(x\)-axis and the vertical \(y\)-axis. The point where the axes intersect is called the origin, denoted by the coordinates \((0, 0)\). Each point on the coordinate plane corresponds to an ordered pair \((x, y)\) that defines its exact location.

In coordinate geometry:

• Points are represented by their \(x\) and \(y\) coordinates.
• Lines can be defined and analyzed using equations.
• The slope of a line is one of the most fundamental concepts, as it determines the line's angle and direction.

By using coordinate geometry, one can not only calculate the slope of a line but also draw relationships and solve real-life problems involving various geometric shapes.

Linear Equations

Linear equations form the foundation of understanding relations and patterns in algebra and coordinate geometry. A linear equation is an algebraic expression that represents a straight line when plotted on a graph.

These equations are typically written in the form \(y = mx + b\), where:

• \(y\) is the dependent variable, representing the line's output value.
• \(x\) is the independent variable, representing the input value.
• \(m\) is the slope of the line, indicating how steep the line is.
• \(b\) is the y-intercept, the point where the line crosses the \(y\)-axis.

Because of their straightforward nature, linear equations make it easy to predict and analyze the relationship between variables.

They are used extensively in various fields, such as economics, physics, and any other area that requires modeling relationships in a linear fashion. Understanding linear equations helps students tackle more complex mathematical concepts and demonstrates how algebra is applied in real-world scenarios.