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Chapter 3: Problem 38

Graph each linear equation. See Examples 6 through \(10 .\) $$ 4=x-3 y $$

Short Answer

Expert verified
Rearrange to slope-intercept form, plot y-intercept at (0,-4/3), use slope 1/3 to plot second point, draw line.

Step by step solution

01

Rearrange the Equation

Start by rearranging the given linear equation into the slope-intercept form, which is \( y = mx + b \). We have the equation \( 4 = x - 3y \). First, we need to solve for \( y \).Subtract \( x \) from both sides: \[ 4 - x = -3y \] Divide both sides by \(-3\) to solve for \( y \): \[ y = \frac{x - 4}{3} \].This is now in the slope-intercept form, where \( m = \frac{1}{3} \) and \( b = -\frac{4}{3} \).
02

Identify the Slope and Y-Intercept

From the rearranged equation \( y = \frac{1}{3}x - \frac{4}{3} \), identify the slope \( m = \frac{1}{3} \) and the y-intercept \( b = -\frac{4}{3} \). This means the line crosses the y-axis at the point \( (0, -\frac{4}{3}) \).
03

Plot the Y-Intercept

On a coordinate plane, plot the point \( (0, -\frac{4}{3}) \), which is the y-intercept of the line. This is the starting point for graphing the line.
04

Use the Slope to Find Another Point

The slope \( \frac{1}{3} \) means that for every 3 units you move horizontally to the right, you move 1 unit vertically up. From the y-intercept \( (0, -\frac{4}{3}) \), move 3 units to the right to get to the point \( (3, -\frac{4}{3}) \), and then 1 unit up to \( (3, \frac{-4}{3} + 1) = (3, -\frac{1}{3}) \). Plot this second point.
05

Draw the Line

With the points \((0, -\frac{4}{3})\) and \((3, -\frac{1}{3})\) plotted on the graph, use a ruler to draw a straight line passing through these points. This line represents the graph of the equation \( 4 = x - 3y \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
When dealing with linear equations, one of the most useful formats to use is the slope-intercept form. This format expresses a linear equation as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. The equation in this form makes graphing straightforward because it directly reveals the slope and y-intercept.

In practical terms, the slope-intercept form allows you to immediately identify the starting point on the y-axis (given by \( b \)) and how steep or flat the line is (indicated by \( m \)). If you are transforming or rearranging an equation, always aim to express it in this form when planning to graph it on a coordinate plane.
Slope
The slope is a measure of how steep a line is and is often referred to as the "rise over run". In the equation \( y = mx + b \), the slope is represented by \( m \). Specifically, \( m = \frac{\text{rise}}{\text{run}} \), which indicates how much the line goes up or down vertically (rise) for a specific horizontal movement (run).

If the slope is positive, the line slants upwards as it moves to the right. Conversely, a negative slope means the line slants downwards. A slope of zero represents a horizontal line.
  • If \( m = \frac{1}{3} \), this means for every 3 units moved to the right, the line moves 1 unit up.
  • Understanding the slope is crucial because it affects the angle and direction of your line.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is a vital component when graphing, serving as the initial anchor point of your line. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \).

For instance, if \( b = -\frac{4}{3} \), this means the line intersects the y-axis at the point \( (0, -\frac{4}{3}) \). To start graphing a linear equation, plot the y-intercept on your graph first. From there, you can use the slope to determine the next points to plot and draw your line through these points.
  • Remember: the y-intercept provides the starting location of the line on the graph.
  • It is found directly by analyzing the equation's constant term in the slope-intercept form.

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