Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the slope and y-intercept (if possible) of the linear equation. Then describe its graph. $$3 y+2=0$$

Step-by-Step Solution

Answer

The slope of the linear equation \(3y+2=0\) is \(0\) and the y-intercept is \(-2/3\). Its graph is a straight horizontal line, lying below the x-axis at \(y=-2/3\).

Step 1: Rewrite the Equation in Slope-Intercept Form

First, rewrite \(3y+2=0\) to the form \(y=mx+c\). Subtract 2 from both sides to isolate \(3y\) on one side of the equation. We get \(3y=-2\). Then, divide the entire equation by 3 to solve for y: \(y=-2/3\). Now, we have \(y=0*x-2/3\). This tells us that the slope \(m=0\) and the y-intercept \(c=-2/3\).

Step 2: Describe the graph

The graph of any linear equation in the form \(y=mx+c\) is a straight line. Here, our slope \(m=0\) means that the line is horizontal. It crosses the y-axis at the point \(0,-2/3\), which is our y-intercept \(c=-2/3\). So, the graph is a straight horizontal line, lying below the x-axis at \(y=-2/3\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing a line using the formula \( y = mx + c \). In this formula, \( y \) denotes the dependent variable, \( x \) is the independent variable, \( m \) represents the slope of the line, and \( c \) is the y-intercept.
This form makes it very easy to understand the characteristics of the line graphically.
For example, by directly looking at the formula, we can determine both the slope and y-intercept of the line.

• The slope \( m \) indicates the steepness and direction of the line.
• The y-intercept \( c \) is the point where the line crosses the y-axis.

In the given problem, we rearranged the equation \( 3y + 2 = 0 \) into the slope-intercept form. After rearranging, it became \( y = 0 \, x - \frac{2}{3} \). Here, you can directly see that the slope \( m \) is 0 and the y-intercept \( c \) is \(-\frac{2}{3}\). This setup will help you to easily graph the line.

Y-intercept

The y-intercept is a crucial element in understanding the graph of a linear equation. It is the exact point at which the line crosses the y-axis. In simpler terms, it is the value of \( y \) when \( x \) is zero. This makes it an easily identifiable feature on a graph.
In our exercise, once the original equation was transformed into the slope-intercept form as \( y = 0 \, x - \frac{2}{3} \), we could pinpoint the y-intercept as \( -\frac{2}{3} \).

• This means that when \( x = 0 \), \( y \) will always be \(-\frac{2}{3}\).
• The line of the equation crosses the y-axis at this point.

The understanding of the y-intercept can help to locate where the line starts on a graph.

Horizontal Line

A horizontal line on a graph corresponds to a situation where the value of \( y \) remains constant no matter what \( x \) is. Unlike diagonal lines, horizontal lines have a slope of zero, meaning there is no vertical change as \( x \) increases or decreases.
In your exercise, the equation \( y = -\frac{2}{3} \) indicates a horizontal line parallel to the x-axis.

• The slope \( m = 0 \), confirming that the line is perfectly flat.
• The y-value \(-\frac{2}{3}\) stays constant across all points on the line.

Horizontal lines are easy to spot in graphs due to their lack of incline, reflecting the same \( y \)-value everywhere on the graph..

Graph Description

Describing a graph involves understanding the appearance and behavior of the line that represents the linear equation. For a linear equation in the slope-intercept form like \( y = mx + c \), the key characteristics can be directly described by the values of \( m \) and \( c \).
In our scenario, based on the equation \( y = -\frac{2}{3} \):

• The slope \( m = 0 \) leads us to conclude the line is horizontal.
• The y-intercept \( c = -\frac{2}{3} \) means the line crosses the y-axis at this point.
• The line is entirely below the x-axis given that the y-intercept is negative.

This explanation helps create a mental image of the graph as a flat line running parallel to the x-axis, cutting the y-axis exactly at \( y = -\frac{2}{3} \). Understanding these characteristics allows one to graph the equation accurately without plotting numerous points.