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Chapter 2: Problem 10

First, graph the equation and determine visually whether it is symmetric with respect to the \(x\) -axis, the \(y\) -axis, and the origin. Then verify your assertion algebraically. $$2 x-5=3 y$$

Short Answer

Expert verified
The graph is symmetric with respect to none of the axes or the origin.

Step by step solution

01

Solve for y

To make the equation easier to graph, solve for y in terms of x. Start with the given equation\[2x - 5 = 3y\]Rearrange it to solve for y:\[y = \frac{2x - 5}{3}\]
02

Graph the Equation

Plot the equation \(y = \frac{2x - 5}{3}\) on a coordinate plane. Identify the y-intercept as \(\left(0, -\frac{5}{3}\right)\) and use the slope \(\frac{2}{3}\) to plot additional points.
03

Check Symmetry Visually

Analyze the graph: - If the graph mirrors itself across the x-axis, it is symmetric with respect to the x-axis. - If the graph mirrors itself across the y-axis, it is symmetric with respect to the y-axis. - If the graph looks the same rotated 180 degrees about the origin, it is symmetric with respect to the origin.
04

Check Symmetry Algebraically for x-axis

To check for symmetry with respect to the x-axis, replace y with -y in the equation \[2x - 5 = 3(-y)\] This yields \[2x - 5 = -3y \Rightarrow y = -\frac{2x - 5}{3}\] This is not the same equation as the original, so the graph is not symmetric with respect to the x-axis.
05

Check Symmetry Algebraically for y-axis

To check for symmetry with respect to the y-axis, replace x with -x in the equation \[2(-x) - 5 = 3y\] This yields \[-2x - 5 = 3y \Rightarrow y = \frac{-2x - 5}{3}\] This is not the same equation as the original, so the graph is not symmetric with respect to the y-axis.
06

Check Symmetry Algebraically for Origin

To check for symmetry with respect to the origin, replace x with -x and y with -y in the equation \[2(-x) - 5 = 3(-y)\] This yields \[-2x - 5 = -3y \Rightarrow y = \frac{-2x - 5}{3}\] This is not the same equation as the original, so the graph is not symmetric with respect to the origin.

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Key Concepts

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

slope-intercept form
The slope-intercept form is a way to write the equation of a line so it's easier to graph. The general format is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
For the equation \(2x - 5 = 3y\), we solve for \(y\) to get it into slope-intercept form: \[y = \frac{2x - 5}{3}\]
Here, the slope \(m\) is \(\frac{2}{3}\) and the y-intercept \(b\) is \(-\frac{5}{3}\). Plot the y-intercept on the y-axis and use the slope to find other points.
x-axis symmetry
A graph is symmetric with respect to the x-axis if it looks the same on both sides of the x-axis.
To test this algebraically, replace \(y\) with \(-y\) in the equation and see if you get the original equation back.
For the equation \[2x - 5 = 3(-y)\], simplifying gives \(-3y = 2x - 5 <=> y = -\frac{2x - 5}{3}\).
This is not the same as the original equation, so the graph is not symmetric with respect to the x-axis.
y-axis symmetry
A graph is symmetric with respect to the y-axis if it mirrors itself on both sides of the y-axis.
To verify this algebraically, replace \(x\) with \(-x\) and see if you get the original equation.
For the equation \[2(-x) - 5 = 3y\], when simplified, this becomes \<-3y = -2x - 5 <=> y = \frac{-2x - 5}{3}.This is not the original equation, so the graph does not have y-axis symmetry.
origin symmetry
A graph is symmetric with respect to the origin if rotating it 180 degrees results in the same graph.
Algebraically, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation.For the equation \[2(-x) - 5 = 3(-y)\], when simplified, this results in:
yavelength: \[y = \frac{-2x -5}{3}\].This is still not the original equation, so the graph is not symmetric with respect to the origin.

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