Problem 57 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. $$ (-3,2),(-4,6) $$

Step-by-Step Solution

Answer

Answer: -4

Step 1: Identify the coordinates of the points

The given points are \((-3, 2)\) and \((-4, 6)\). We can label these as follows: $$ (x1, y1) = (-3, 2) \\ (x2, y2) = (-4, 6) $$

Step 2: Apply the slope formula

Using the formula for finding the slope \(m\), we can plug in the coordinates of the points: $$ m = \frac{6 - 2}{-4 - (-3)} $$

Step 3: Simplify the expression

Now, we will simplify the expression to find the slope: $$ m = \frac{4}{-1} $$

Step 4: Find the slope

After simplifying, we find that the slope \(m\) is: $$ m = -4 $$ So, the slope of the line through the given points is \(-4\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols, when put together, form mathematical expressions which can represent both numbers and relationships. In the context of finding the slope, algebra provides a systematic way of calculating the rate at which one variable changes with respect to another.

For instance, when evaluating the slope between two points, we leverage algebraic expressions to represent the change in the 'y' values over the change in the 'x' values. By simplifying these expressions, we can decode the nature of the line connecting our points - whether it's increasing, decreasing, constant, or undefined. Mastering algebraic techniques is essential for understanding linear relationships and how they are graphically represented in coordinate geometry.

Slope Formula

The slope formula is a critical tool in coordinate geometry for analyzing linear relationships between points on a Cartesian plane. It is symbolically represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( m \) stands for the slope of the line, \( (x_1,y_1) \) are the coordinates of the first point, and \( (x_2,y_2) \) are the coordinates of the second point.

The formula essentially gives us the 'steepness' or 'gradient' of a line. A positive slope suggests an upward tilt from left to right, while a negative slope indicates a downward tilt. A slope of zero signifies a perfectly horizontal line. For vertical lines, where the change in x is zero, the slope is undefined. Understanding how to apply the slope formula and interpret its result is foundational for working with linear equations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through the use of coordinates and algebraic principles. This branch of mathematics allows us to represent points, lines, and shapes in a two-dimensional plane using ordered pairs like \( (x, y) \).

When finding the slope of the line through two points, coordinate geometry is at the forefront. It gives us a visual perspective on what the algebraic manipulation of the slope formula represents. For example, plotting the points on a graph can help us visualize their relationship and the slope as a measure of the line's steepness, whether the line goes up or down and how quickly. This visualization is a powerful tool in understanding the nature of linear equations and the way they model real-world relationships.

Linear Equations

Linear equations form the backbone of linear relationships represented in both algebra and coordinate geometry. A linear equation can be recognized by its standard form \( Ax + By = C \), or its slope-intercept form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The graph of a linear equation is always a straight line.

A core aspect of linear equations is the concept of slope, which indicates the direction and the rate at which the value of \( y \) changes with respect to \( x \). Understanding how to find and interpret the slope is key for solving problems involving linear equations, such as predicting trends and analyzing patterns. It helps in constructing graphs and modeling various real-life scenarios, such as calculating velocity or economic growth.