Chapter 1, Problem 112RE

To determine

To graph: The equation $2x-y+1=0$ and check the equation for symmetry.

Answer to Problem 112RE

The equation $2x-y+1=0$ is not symmetric and the graph is as follows:

  

Explanation of Solution

Given information:

The equation of line $2x-y+1=0$

Formula used:

The equation is said to have symmetry about the $x$ -axis when the equation remains unchanged on replacing $y$ by $-y$

Graphically it can be said that the graph remains unchanged when reflected in the $x$ -axis.

The equation is said to have symmetry about the $y$ -axis when the equation remains unchanged on replacing $x$ by $-x$

Graphically it can be said that the graph remains unchanged when reflected in the $y$ -axis.

The equation is said to have symmetry about the origin if it is symmetric about $x$ -axis and $y$ -axis that is the equation remains unchanged on replacing $x$ by $-x$ or $y$ by $-y$

Graphically it can be said that the graph remains unchanged when rotated with an angle of ${180}^{\circ }$ about the origin.

Calculation:

It is provided that the equation of line is $2x-y+1=0$ $\left(1\right)$

Firstly replace $y$ by $-y$ in $\left(1\right)$

  $\begin{array}{l}2x-\left(-y\right)+1=0\\ 2x+y+1=0\end{array}$

The equation obtained is not the same as the original one.

Hence, $2x-y+1=0$ is not symmetric about $x$ -axis

Next, replace $x$ by $-x$ in $\left(1\right)$

  $\begin{array}{l}2\left(-x\right)-y+1=0\\ -2x-y+1=0\\ 2x+y-1=0\end{array}$

The equation obtained is not the same as the original one.

Hence, $2x-y+1=0$ is not symmetric about $y$ -axis.

Thus, $2x-y+1=0$ is not symmetric about origin.

For plotting:

$x$	$2x-y+1=0$	$\left(x.y\right)$
$\begin{array}{l}0\\ 1\\ 2\end{array}$	$\begin{array}{l}1\\ 3\\ 5\end{array}$	$\begin{array}{l}\left(0,1\right)\\ \left(1,3\right)\\ \left(2,5\right)\end{array}$

We plot these points to get the graph as follows:

  

Thus, $2x-y+1=0$ is neither symmetric about $x$ -axis nor about $y$ -axis.