Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, find the slope of the line that passes through the two given points. (-1,4) and (5,2)

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Points

We start by identifying the points given in the exercise. The two points are (-1,4) and (5,2). We will use these points to find the slope of the line.

Step 2: Recall the Slope Formula

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

Step 3: Substitute Points into the Formula

Substitute the given points into the slope formula. Assign \(x_1 = -1\), \(y_1 = 4\), \(x_2 = 5\), and \(y_2 = 2\). The formula becomes: \[ m = \frac{2 - 4}{5 - (-1)} \]

Step 4: Calculate the Differences

Calculate the difference in the y-values and the x-values: - \(2 - 4 = -2\) - \(5 - (-1) = 5 + 1 = 6\).

Step 5: Simplify the Slope

Substitute the calculated differences back into the formula: \[ m = \frac{-2}{6} \]. Simplify the fraction by dividing both the numerator and the denominator by 2, resulting in \[ m = \frac{-1}{3} \].

Key Concepts

Slope Formula

Understanding the slope formula is essential when dealing with lines in coordinate geometry. Slope is a measure of the steepness of a line, and it's calculated using the difference in the y-coordinates over the difference in the x-coordinates of two distinct points on the line. The formula for slope is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where

• \(m\) denotes the slope,
• \((x_1, y_1)\) are the coordinates of the first point,
• \((x_2, y_2)\) are the coordinates of the second point.

Using this formula, we can determine the steepness and direction of a line. A positive slope indicates an upward trend from left to right, while a negative slope shows a downward trend. If the slope is zero, the line is horizontal. A more vertical line will have a greater slope in absolute terms.
Moreover, the formula assumes that the points are different; otherwise, the slope will be undefined due to division by zero. This formula forms the basis for analyzing linear relationships and is a crucial part of coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method allows us to solve geometric problems algebraically and vice versa.
Points in coordinate geometry are represented as ordered pairs \((x, y)\). These pairs denote positions on the Cartesian plane, which is a two-dimensional plane with horizontal (x-axis) and vertical (y-axis) lines.
Using coordinate geometry, we can:

• Find the distance between two points using the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
• Determine the midpoint of a segment connecting two points with the formula \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\).
• Explore the properties of figures like triangles, rectangles, or circles.

Coordinate geometry provides a powerful toolset for analyzing and solving problems that were otherwise purely geometric. This consistent method bridges algebra with geometry, allowing for more comprehensive problem-solving techniques.

Linear Equations

Linear equations are fundamental in mathematics and describe lines in the coordinate plane. They are expressions that model linear relationships between variables, usually in the form: \[ y = mx + c \] where

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

Linear equations can take different forms, such as standard form \(Ax + By = C\). They are "linear" because they graph as straight lines. By understanding the structure of a linear equation, one can easily graph it and understand the role of both the slope \(m\) and intercept \(c\).
For example, increasing the value of \(m\) makes the line steeper, while adjusting \(c\) shifts the line up or down. Linear equations are key in modeling real-world scenarios, such as calculating speed, representing financial data, or predicting outcomes. They form a foundational concept that students will encounter across various mathematical contexts.