Problem 101 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ for the given ordered pairs \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). $$\left(x_{1}, y_{1}\right)=(1,3) ;\left(x_{2}, y_{2}\right)=(6,13)$$

Step-by-Step Solution

Answer

The slope of the line through the points (1,3) and (6,13) is 2.

Step 1: Understand the problem and the given

The problem is asking to find the slope of the line through two given points \((x_1, y_1) = (1,3)\) and \((x_2, y_2) = (6,13)\).

Step 2: Using the formula of the slope

The formula of the slope is given by \( \frac{y_2 - y_1}{x_2 - x_1} \). We simply need to fill in the formula with the right values from the two ordered pairs.

Step 3: Substitute and solve

Substitute \(x_1=1\), \(y_1=3\), \(x_2=6\), \(y_2=13\) into the formula: \( \frac{13-3}{6-1} = \frac{10}{5} = 2 \).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. As an integral part of algebra, understanding the slope of a line requires us to delve into variables and operations. The slope itself is a measure of steppness and direction of a straight line on a graph.

When trying to find the slope of a line through two points, as in the textbook exercise, you will use the algebraic formula \( \frac{y_2 - y_1}{x_2 - x_1} \). In essence, you're calculating the 'rise over run', or how much the line goes up or down (rise) for a given movement to the right (run) on the coordinate plane. The result helps in understanding the behavior of linear relationships in algebraic problems.

Linear Equations

Linear equations are foundational in the study of algebra. They can be visualized as straight lines when plotted on a graph and are presented in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Slope is the rate of change between any two points on the line, and in the given exercise, we assess the slope by evaluating \( \frac{y_2 - y_1}{x_2 - x_1} \).

By determining the slope of the line that passes through two given points, \( (1,3) \) and \( (6,13) \) in this case, we are finding the 'm' in the linear equation. Once we have the slope, we have a critical piece to understand the direction and steepness of our line, as well as being able to predict other points on the line or drafting the line on a graph.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It allows us to analyze geometrical shapes and relationships between them using algebraic equations. The clear illustration here is the concept of the slope, which quantifies how a line inclines away from the horizontal axis.

In coordinate geometry, plotting the points \( (1,3) \) and \( (6,13) \) on the Cartesian plane allows us to visually grasp the slope. Connecting these dots results in a straight line, and applying our algebraic knowledge to compute the rise over run (slope), we see a practical example of the intersection of algebra and geometry. The slope value obtained is not just a number but a geometric property that can indicate whether the line is rising or falling as it moves from left to right across the graph. This merger of algebraic methods and geometric figures is a cornerstone concept in coordinate geometry.