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Chapter 3: Problem 62

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 x=9 y $$

Short Answer

Expert verified
A line passing through the origin with a slope of \(\frac{1}{3}\).

Step by step solution

01

Convert the Equation to Slope-Intercept Form

The given equation is \(3x = 9y\). We need to convert this to the slope-intercept form, \(y = mx + b\). Divide both sides by 9 to isolate y: \[\frac{3x}{9} = \frac{9y}{9} \ Rightarrow y = \frac{1}{3}x.\]
02

Identify the Slope and Y-Intercept

The equation is now in the form \(y = mx + b\), where m is the slope and b is the y-intercept. Here, \(m = \frac{1}{3}\) and \(b = 0\). So, the slope is \(\frac{1}{3}\) and the y-intercept is 0.
03

Graph the Y-Intercept

Start by plotting the y-intercept on the coordinate plane. The y-intercept is 0, meaning the line passes through the origin (0, 0).
04

Use the Slope to Plot Another Point

The slope \(\frac{1}{3}\) means that for every 3 units you move horizontally, you move 1 unit vertically upwards. From the origin (0, 0), move 3 units to the right to (3, 0) and 1 unit up to (3, 1).
05

Draw the Line

With the points (0, 0) and (3, 1), draw a straight line through these points. This line represents the graph of the equation \(3x = 9y\).

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Key Concepts

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

slope-intercept form
In mathematics, the slope-intercept form is commonly used to represent linear equations. The general form is given by \(y = mx + b\), where 'm' represents the slope of the line and 'b' is the y-intercept.
The slope 'm' determines the steepness and direction of the line. A positive slope means the line ascends from left to right, while a negative slope means it descends. The y-intercept 'b' indicates where the line crosses the y-axis.
For example, in the equation \(3x = 9y\), we can convert it to slope-intercept form by solving for y:
  • Divide both sides by 9: \(\frac{3x}{9} = y \rightarrow y = \frac{1}{3}x\)

Now, we have \(y = \frac{1}{3}x\), which is in the form \(y = mx + b\), where \(m = \frac{1}{3}\) and \(b = 0\).
slope and y-intercept
To graph a linear equation, it's essential to understand the slope and y-intercept.
The **slope (m)** describes how the y-value changes with respect to the x-value. If \(m > 0\), the line rises as it moves from left to right. If \(m < 0\), the line falls as it moves from left to right. The **y-intercept (b)** is where the line crosses the y-axis.
In our example, the equation \(y = \frac{1}{3}x\) reveals:
  • Slope (m): \(\frac{1}{3}\)
  • Y-Intercept (b): 0
This tells us the line crosses the y-axis at (0,0) and rises gradually, going up 1 unit for every 3 units it moves to the right.
coordinate plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
We use the coordinate plane to graph equations by plotting points that satisfy the equation.
For the equation \(3x = 9y\) converted to \(y = \frac{1}{3}x\):
  • First, plot the y-intercept (0,0)
  • Then, use the slope (\(\frac{1}{3}\)): from (0,0), move 3 units to the right and 1 unit up to reach (3,1)
  • Draw a straight line through these points
These steps will help you accurately graph any linear equation on the coordinate plane.

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Most popular questions from this chapter

A function defined by the equation of a line, such as $$ f(x)=3 x-4 $$ is called a linear function. It can be graphed by replacing \(f(x)\) with \(y\) and then using the methods described earlier in this chapter. Let us assume that some function is written in the form \(f(x)=m x+b,\) for particular values of \(m\) and \(b .\) If \(f(2)=4,\) name the coordinates of one point on the line.

Graph each linear equation. \(x=y+2\)

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 x=y-9 $$

Solve each equation. $$ 4+2 x=10 $$

What is the equation of the \(x\) -axis? What is the equation of the \(y\) -axis?
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