Problem 8 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ x+3 y=5 $$

Step-by-Step Solution

Answer

1. Rewrite as \(y = -\frac{1}{3}x + \frac{5}{3}\). Plot \((0, \frac{5}{3})\) and \((1, \frac{4}{3})\). Draw the line.

Step 1: Rewrite in Slope-Intercept Form

First, let's rewrite the given equation in the form of \(y = mx + c\). To do that, solve for \(y\) in the equation: \(x + 3y = 5\). Subtract \(x\) from both sides to get: \(3y = -x + 5\). Then, divide every term by 3 to isolate \(y\): \(y = -\frac{1}{3}x + \frac{5}{3}\). Now, the equation is in slope-intercept form \(y = mx + c\).

Step 2: Identify Slope and Y-Intercept

In the equation \(y = -\frac{1}{3}x + \frac{5}{3}\), identify the slope \(m\) and the y-intercept \(c\). The slope \(m\) is \(-\frac{1}{3}\), meaning for every 1 unit increase in \(x\), \(y\) decreases by \( \frac{1}{3} \). The y-intercept \(c\) is \(\frac{5}{3}\), which is the point where the line crosses the y-axis.

Step 3: Plot the Y-Intercept

Start graphing by plotting the y-intercept \((0, \frac{5}{3})\) on the graph. This is the starting point of the line.

Step 4: Use the Slope to Find Another Point

From the y-intercept \((0, \frac{5}{3})\), apply the slope \(-\frac{1}{3}\) to find another point. Move 1 unit to the right (positive x-direction), and \(-\frac{1}{3}\) unit down (negative y-direction) to reach the point \((1,\frac{5}{3} - \frac{1}{3}) = (1, \frac{4}{3})\).

Step 5: Draw the Line

With two points \((0, \frac{5}{3})\) and \((1, \frac{4}{3})\), draw a straight line through them extending in both directions, as lines continue infinitely.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a significant tool in understanding linear relationships. It is represented by the equation \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept. This form simplifies the process of graphing a line.
To convert any linear equation to the slope-intercept form, you need to solve for \( y \). For instance, consider the equation \( x + 3y = 5 \). By rearranging the terms and isolating \( y \), we obtain \( y = -\frac{1}{3}x + \frac{5}{3} \). This manipulation brings the equation into an accessible format, making it easier to identify the key components for graphing.

Plotting Points

Plotting points is a core skill that allows you to visually represent equations on a graph. Once a linear equation is in slope-intercept form, plotting can commence with the y-intercept. For the equation \( y = -\frac{1}{3}x + \frac{5}{3} \), the y-intercept is \( \frac{5}{3} \), which translates to the point \( (0, \frac{5}{3}) \) on a graph.
The next step involves using the slope to find additional points. The slope of \( -\frac{1}{3} \) tells you that for every 3 units you move to the right, you move 1 unit down, or equivalently, for every 1 unit to the right, the movement is \( -\frac{1}{3} \) unit down. Starting from \( (0, \frac{5}{3}) \), moving to \( (1, \frac{4}{3}) \) gives the next point on the line. This careful plotting is crucial to maintain the accuracy and representation of the linear relationship.

Identifying Slope and Y-Intercept

Identification of the slope and y-intercept is pivotal in understanding the attributes of a linear graph. The slope \( m \), typically a fraction or a whole number, dictates the direction and steepness of the line. For \( y = -\frac{1}{3}x + \frac{5}{3} \), the slope is \(-\frac{1}{3}\), indicating a gentle downward angle as \( x \) increases.
The y-intercept \( c \) is equally vital as it shows where the line crosses the y-axis. It is the starting point for graphing, here being \( \frac{5}{3} \) on the y-axis (point \( (0, \frac{5}{3}) \)). Together, slope and y-intercept provide a comprehensive blueprint for sketching the line, helping decipher its behavior and intersection with the axes. Understanding these concepts transforms the abstract algebraic expressions into clear graphical representations.