Problem 5 In each of Exercises \(1-6,\) ma... [FREE SOLUTION] (2024)

Short Answer

Linear Equation

A linear equation is a mathematical statement that represents a straight line on a graph. The general form of a linear equation in slope-intercept form is given by the equation \( y = mx + b \). Here, \( y \) and \( x \) are variables, \( m \) represents the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis. Linear equations are fundamental in algebra and are used to model relationships between two variables in various real-life situations.

Calculating Slope

Calculating the slope of a line is an essential skill when working with linear equations. The slope describes the steepness and direction of the line. It is calculated using the formula: \( m = \frac{\text{Difference in } y}{\text{Difference in } x} \). In simpler terms, the slope \( m \) is the ratio of the change in vertical distance (the rise) to the change in horizontal distance (the run) between two points on the line. Understanding how to calculate the slope allows you to determine how much \( y \) changes for a given change in \( x \).

Difference in Variables

When discussing linear equations, it is important to understand the concept of the difference in variables. The difference in \( x \), often denoted as \( \Delta x \), represents the change or distance between two x-values. Similarly, the difference in \( y \), denoted as \( \Delta y \), represents the change or distance between two y-values. These differences are fundamental when calculating the slope of a line. Knowing both \( \Delta x \) and \( \Delta y \) helps you understand the relationship between the two variables, which is crucial for analyzing the behavior of linear functions.

Point on a Graph

A point on a graph is represented by an ordered pair \((x, y)\), showing the position of that point in the Cartesian coordinate system. Each point on a line graphed from a linear equation corresponds to a specific solution to the equation. For example, the y-intercept, given as \((0, b)\), is where the line crosses the y-axis. This means when \( x = 0 \), the value of \( y \) is \( b \). Understanding points on a graph helps in visualizing and sketching the graph of a linear equation. It also aids in interpreting solutions and patterns within real-world data scenarios.