Find the Symmetry (7x)/(x-2)

$\frac{7 x}{x - 2}$

Step 1

Write $\frac{7 x}{x - 2}$ as an equation.

$y = \frac{7 x}{x - 2}$

Step 2

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 3

If $(x, y)$ exists on the graph, then the graph is symmetric about the:

1. X-Axis if $(x, - y)$ exists on the graph

2. Y-Axis if $(- x, y)$ exists on the graph

3. Origin if $(- x, - y)$ exists on the graph

Step 4

Check if the graph is symmetric about the x-axis by plugging in $- y$ for $y$ .

$- y = \frac{7 x}{x - 2}$

Step 5

Simplify the left side.

Remove parentheses.

$- y = \frac{7 x}{x - 2}$

Step 6

Since the equation is not identical to the original equation, it is not symmetric to the x-axis.

Not symmetric to the x-axis

Step 7

Check if the graph is symmetric about the y-axis by plugging in $- x$ for $x$ .

$y = \frac{7 (- x)}{(- x) - 2}$

Step 8

Simplify the right side.

Simplify $\frac{7 (- x)}{(- x) - 2}$ .

Simplify the expression.

Multiply $- 1$ by $7$ .

$y = \frac{- 7 x}{- x - 2}$

Move the negative in front of the fraction.

$y = - \frac{(7) x}{- x - 2}$

Factor $- 1$ out of $- x$ .

$y = - \frac{7 x}{- (x) - 2}$

Rewrite $- 2$ as $- 1 (2)$ .

$y = - \frac{7 x}{- (x) - 1 (2)}$

Factor $- 1$ out of $- (x) - 1 (2)$ .

$y = - \frac{7 x}{- (x + 2)}$

Simplify the expression.

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$y = - \frac{7 x}{- 1 (x + 2)}$

Move the negative in front of the fraction.

$y = - - \frac{7 x}{x + 2}$

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{7 x}{x + 2}$

Multiply $\frac{7 x}{x + 2}$ by $1$ .

$y = \frac{7 x}{x + 2}$

Step 9

Since the equation is not identical to the original equation, it is not symmetric to the y-axis.

Not symmetric to the y-axis

Step 10

Check if the graph is symmetric about the origin by plugging in $- x$ for $x$ and $- y$ for $y$ .

$- y = \frac{7 (- x)}{(- x) - 2}$

Step 11

Solve for $y$ .

Simplify the left side.

Remove parentheses.

$- y = \frac{7 (- x)}{(- x) - 2}$

Simplify the right side.

Simplify $\frac{7 (- x)}{(- x) - 2}$ .

Simplify the expression.

Multiply $- 1$ by $7$ .

$- y = \frac{- 7 x}{- x - 2}$

Move the negative in front of the fraction.

$- y = - \frac{(7) x}{- x - 2}$

Factor $- 1$ out of $- x$ .

$- y = - \frac{7 x}{- (x) - 2}$

Rewrite $- 2$ as $- 1 (2)$ .

$- y = - \frac{7 x}{- (x) - 1 (2)}$

Factor $- 1$ out of $- (x) - 1 (2)$ .

$- y = - \frac{7 x}{- (x + 2)}$

Simplify the expression.

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$- y = - \frac{7 x}{- 1 (x + 2)}$

Move the negative in front of the fraction.

$- y = - - \frac{7 x}{x + 2}$

Multiply $- 1$ by $- 1$ .

$- y = 1 \frac{7 x}{x + 2}$

Multiply $\frac{7 x}{x + 2}$ by $1$ .

$- y = \frac{7 x}{x + 2}$

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

$1 y = - (\frac{7 (- x)}{(- x) - 2})$

Multiply $y$ by $1$ .

$y = - (\frac{7 (- x)}{(- x) - 2})$

Simplify the right side.

Simplify $- (\frac{7 (- x)}{(- x) - 2})$ .

Multiply $- 1$ by $7$ .

$y = - \frac{- 7 x}{- x - 2}$

Move the negative in front of the fraction.

$y = - - \frac{(7) x}{- x - 2}$

Multiply $- - \frac{7 x}{- x - 2}$ .

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{7 x}{- x - 2}$

Multiply $\frac{7 x}{- x - 2}$ by $1$ .

$y = \frac{7 x}{- x - 2}$

Factor $- 1$ out of $- x$ .

$y = \frac{7 x}{- (x) - 2}$

Rewrite $- 2$ as $- 1 (2)$ .

$y = \frac{7 x}{- (x) - 1 (2)}$

Factor $- 1$ out of $- (x) - 1 (2)$ .

$y = \frac{7 x}{- (x + 2)}$

Rewrite negatives.

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$y = \frac{7 x}{- 1 (x + 2)}$

Move the negative in front of the fraction.

$y = - \frac{7 x}{x + 2}$

Step 12

Since the equation is not identical to the original equation, it is not symmetric to the origin.

Not symmetric to the origin

Step 13

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 14

Find the Symmetry (7x)/(x-2)

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