Problem 1 - College Algebra Essentials

Question

Find the slope of the line passing through each pair of points or state that the slope is undefined. Then indicate whether the line through the points rises falls, is horizontal, or is vertical. $$(4,7) \text { and }(8,10)$$

Step-by-Step Solution

Answer

The slope of the line passing through points (4,7) and (8,10) is \( \frac{3}{4} \). Since the slope is positive, the line rises.

Step 1: Identify the coordinates

Identify the ordered pairs (x1, y1) and (x2, y2). Here, (x1, y1) = (4,7) and (x2, y2) = (8,10)

Step 2: Calculate the slope

Plug the coordinates into the slope formula, which is \( m = \frac{y2 - y1}{x2 - x1} \). Our slope \( m = \frac{10 - 7}{8 - 4} = \frac{3}{4} \)

Step 3: Determine the characteristic of the line

Since the slope is positive, the line rises.

Key Concepts

Linear Equations

Linear equations form the foundation of algebra and help us find relationships between variables. A linear equation is an equation that forms a straight line when graphed on a coordinate plane. These equations can often be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is known as the slope-intercept form.

Linear equations are valuable because they appear in various real-life situations. They can be used to model anything from simple predictions like budgeting to more complex systems like velocity calculations. Understanding linear equations allows us to predict outcomes, find optimal solutions, and interpret trends effectively.

In the exercise, finding the slope is a part of understanding the linear equation representing the line connecting the points. Once you know the slope and any point on the line, you can express the complete equation of the line.

Coordinate Geometry

Coordinate geometry, or analytic geometry, provides a bridge between algebra and geometry. It uses a coordinate plane to represent equations and geometric figures as algebraic expressions. This branch of mathematics allows us to describe the position of points, lines, and figures in a plane using pairs of numbers, called coordinates.

The coordinate plane consists of two axes: the horizontal x-axis and the vertical y-axis. Each point on this plane is defined by an ordered pair \( (x, y) \). Understanding how to read and plot these coordinates on a grid is essential in solving problems related to the position and shape of geometric figures.

In the given exercise, the ordered pairs \( (4,7) \) and \( (8,10) \) are used to find the slope of the line passing through them. This process highlights the use of coordinate geometry to solve problems involving line characteristics.

Graphical Representation

Graphical representation is a powerful tool for visualizing mathematical concepts. It involves plotting points, lines, and curves on a graph to illustrate relationships between variables. When dealing with linear equations, graphs allow us to see the direction and steepness of a line, known as its slope.

The slope of a line indicates how much the line ascends or descends as it moves from left to right across the graph. A positive slope means that the line rises, while a negative slope suggests it falls. A zero slope results in a horizontal line, and an undefined slope corresponds to a vertical line.

In our exercise, after calculating the slope as \( \frac{3}{4} \), we learned that the line rises. This graphical understanding aids in visually interpreting algebraic results and showcasing data trends, enhancing our problem-solving capabilities.

Algebraic Calculation

Algebraic calculation involves manipulating numbers and variables to derive solutions to problems. Calculating the slope of a line using algebraic methods is a critical step in understanding linear relationships.

The slope \( m \) is found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are points on the line. Substituting values into this formula allows us to calculate the rate at which one variable changes concerning another.

In the provided problem, by substituting the given points \((4,7)\) and \((8,10)\) into the formula, we found a slope of \( \frac{3}{4} \). This step not only gives us the slope but also helps identify characteristics of the line as either rising or falling, based on whether the slope is positive or negative. This kind of algebraic manipulation is essential in many fields, from engineering to economics.