Problem 28 - Beginning and Intermediate Algebra

Question

Use the slope formula to find the slope of the line containing each pair of points. $$(0,4) \text { and }(8,-2)$$

Step-by-Step Solution

Answer

Using the slope formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\) for the points \((0,4)\) and \((8,-2)\), we find that \(m = \frac{{-2 - 4}}{{8 - 0}} = \frac{{-6}}{{8}} = -0.75\). Thus, the slope of the line passing through these points is -0.75.

Step 1: Identify the Coordinates

First, name the coordinates. This will aid in putting them correctly in the slope formula. In our case, \((0,4)\) will be \((x_1, y_1)\) and \((8,-2)\) will be \((x_2, y_2)\).

Step 2: Substitute into the Slope Formula

Next, substitute the identified values into the slope formula. This means, replace \(x_1\), \(y_1\), \(x_2\), and \(y_2\) in the slope formula \((m = \frac{{y_2 - y_1}}{{x_2 - x_1}})\) with 0, 4, 8, and -2 respectively.

Step 3: Carry out the Calculations

After substitution, we have: \[ m = \frac{{-2 - 4}}{{8 - 0}} \] This will simplify to: \[ m = \frac{{-6}}{{8}} \] Diving -6 by 8 gives: \[ m = -0.75 \]

Step 4: Answer

Therefore, the slope of the line passing through the points \((0,4)\) and \((8,-2)\) is -0.75.

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is the study of geometry using a coordinate system. This allows us to describe the positions and shapes of geometric objects algebraically, making complex problems more intuitive and easier to calculate. In the context of lines, we define each point by a pair of numerical coordinates, usually referred to as \(x\) and \(y\). These coordinates represent positions on the Cartesian plane.

The Cartesian plane is a two-dimensional plane defined by two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical). Each point on this plane can be described by its \(x\)-coordinate (horizontal distance from the origin) and its \(y\)-coordinate (vertical distance from the origin). Using coordinate geometry, we can not only identify points such as \( (0, 4) \) and \( (8, -2) \) but also analyze and understand the lines connecting them. This is achieved through concepts such as slope, distance, and the midpoint formula.

Linear Equations

Linear equations are mathematical expressions that define straight lines in geometry. These equations express a relationship between two variables, typically x and y, where the degree of the equation is one.

A basic form of a linear equation is \(y = mx + c\), where:

• \(m\) represents the slope of the line, indicating the steepness and direction.
• \(c\) is the y-intercept, where the line crosses the y-axis.

Linear equations are fundamental in understanding how two quantities vary together, providing a straightforward method to determine points that lie on a particular line in the coordinate plane. When given two points, as in our example, linear equations allow us to not only find the line's slope but also to explore its graphical behavior.

Slope Formula

The slope formula is a method used in coordinate geometry to calculate the steepness of a line defined by two distinct points. The slope \(m\) of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as:\[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]This formula measures the change in the y-coordinates relative to the change in the x-coordinates between the two points, giving rise to what's often described as 'rise over run'.

Let's apply this to our points \( (0, 4) \) and \( (8, -2) \):

• Identify coordinates: \( (x_1, y_1) = (0, 4) \) and \( (x_2, y_2) = (8, -2) \).
• Substitute these into the formula: \[m = \frac{{-2 - 4}}{{8 - 0}} = \frac{{-6}}{{8}} = -0.75\]

The slope of -0.75 indicates the line is decreasing, with a downward incline from left to right. Understanding and using the slope formula is essential for analyzing and interpreting linear relationships in coordinate geometry.