Problem 43 - Algebra and Trigonometry

Question

Find the slope and \(y\)-intercept of the line and draw its graph. \(x+3 y=0\)

Step-by-Step Solution

Answer

Slope is \(-\frac{1}{3}\) and \(y\)-intercept is 0.

Step 1: Convert to Slope-Intercept Form

First, rearrange the equation into the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Start with the original equation:\[ x + 3y = 0 \]Subtract \(x\) from both sides:\[ 3y = -x \]Next, divide every term by 3 to solve for \(y\):\[ y = -\frac{1}{3}x \]

Step 2: Identify the Slope and Y-intercept

In the equation \(y = -\frac{1}{3}x\), compare with \(y = mx + b\) to find the slope \(m\) and \(y\)-intercept \(b\).- Slope \(m = -\frac{1}{3}\)- \(y\)-intercept \(b = 0\) (since there is no constant term).

Step 3: Draw the Graph

To graph the equation, start by plotting the y-intercept (0,0) on the graph since \(b = 0\).Next, use the slope to determine the rise over run. The slope \(-\frac{1}{3}\) means for every 3 units you move horizontally to the right, you move 1 unit downwards.Start from (0,0):1. Move 3 units to the right to (3, 0).2. Then move 1 unit down to (3, -1).Draw the line through these points.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most straightforward ways to express a line. It is written as \( y = mx + b \). In this equation:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

This form is highly preferred because it allows us to quickly understand and graph the behavior of the line. To convert any linear equation into the slope-intercept form, you must solve for \( y \), isolating it on one side of the equation. By doing so, you can easily identify both the slope and the y-intercept, which are crucial for graphing. For example, we started with the equation \( x + 3y = 0 \) and converted it to \( y = -\frac{1}{3}x \). This reveals the line's slope and its y-intercept.

Slope

In the context of a line, the slope is a measure of its steepness and direction. The slope \( m \) is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. A positive slope indicates an upward trend as you move from left to right, whereas a negative slope shows a downward trend. When we have the equation \( y = -\frac{1}{3}x \), the slope is \( -\frac{1}{3} \). This tells us:

• The line travels downward as it moves from left to right.
• For every 3 units of horizontal movement to the right, there's a 1 unit decline vertically.

Understanding the slope is essential for predicting how the line is positioned and deciding the direction in which it will extend on a graph.

Y-intercept

The y-intercept of a line is the point on the graph where the line crosses the y-axis. It is represented by \( b \) in the slope-intercept form equation \( y = mx + b \). In practical terms, it's the value of \( y \) when \( x \) is zero.In our specific example with the equation \( y = -\frac{1}{3}x \), there is no constant term, which indicates that the y-intercept \( b \) is 0. This tells us that the line passes through the origin of the graph (0,0), which is an essential anchor point for plotting the line initially. Knowing the y-intercept helps in positioning the line accurately from the start when graphing.

Graphing Linear Equations

Graphing a linear equation involves representing it visually on a coordinate plane. The first step involves identifying the y-intercept and plotting this point. For the equation \( y = -\frac{1}{3}x \), the y-intercept is (0,0). Next, utilize the slope to find another point on the line. Since the slope is \( -\frac{1}{3} \), start at the y-intercept:

• Move 3 units right horizontally (the 'run').
• Then, move 1 unit down vertically (the 'rise').

This gives another point on the line (3, -1). Plot this point and draw a straight line through it and the y-intercept. This process demonstrates how the slope and y-intercept can be used to accurately draw the line, providing a clear picture of the linear relationship represented by the equation.