Algebra Examples

$\frac{{(x - 3)}^{2}}{81} - \frac{{(y + 5)}^{2}}{16} = 1$

Step 1

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 2

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 3

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

$\frac{{(x - 3)}^{2}}{81} - \frac{{(- y + 5)}^{2}}{16} = 1$

Step 4

Simplify.

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Step 4.1

To write $\frac{{(x - 3)}^{2}}{81}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{{(x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{{(- y + 5)}^{2}}{16} = 1$

Step 4.2

To write $- \frac{{(- y + 5)}^{2}}{16}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$\frac{{(x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{{(- y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 4.3

Write each expression with a common denominator of $1296$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{{(x - 3)}^{2}}{81}$ by $\frac{16}{16}$ .

$\frac{{(x - 3)}^{2} \cdot 16}{81 \cdot 16} - \frac{{(- y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 4.3.2

Multiply $81$ by $16$ .

$\frac{{(x - 3)}^{2} \cdot 16}{1296} - \frac{{(- y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 4.3.3

Multiply $\frac{{(- y + 5)}^{2}}{16}$ by $\frac{81}{81}$ .

$\frac{{(x - 3)}^{2} \cdot 16}{1296} - \frac{{(- y + 5)}^{2} \cdot 81}{16 \cdot 81} = 1$

Step 4.3.4

Multiply $16$ by $81$ .

$\frac{{(x - 3)}^{2} \cdot 16}{1296} - \frac{{(- y + 5)}^{2} \cdot 81}{1296} = 1$

Step 4.4

Combine the numerators over the common denominator.

$\frac{{(x - 3)}^{2} \cdot 16 - {(- y + 5)}^{2} \cdot 81}{1296} = 1$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Rewrite ${(x - 3)}^{2} \cdot 16$ as ${((x - 3) \cdot 4)}^{2}$ .

$\frac{{((x - 3) \cdot 4)}^{2} - {(- y + 5)}^{2} \cdot 81}{1296} = 1$

Step 4.5.2

Rewrite ${(- y + 5)}^{2} \cdot 81$ as ${((- y + 5) \cdot 9)}^{2}$ .

$\frac{{((x - 3) \cdot 4)}^{2} - {((- y + 5) \cdot 9)}^{2}}{1296} = 1$

Step 4.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = (x - 3) \cdot 4$ and $b = (- y + 5) \cdot 9$ .

$\frac{((x - 3) \cdot 4 + (- y + 5) \cdot 9) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4

Simplify.

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Step 4.5.4.1

Apply the distributive property.

$\frac{(x \cdot 4 - 3 \cdot 4 + (- y + 5) \cdot 9) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.2

Move $4$ to the left of $x$ .

$\frac{(4 \cdot x - 3 \cdot 4 + (- y + 5) \cdot 9) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.3

Multiply $- 3$ by $4$ .

$\frac{(4 \cdot x - 12 + (- y + 5) \cdot 9) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.4

Apply the distributive property.

$\frac{(4 x - 12 - y \cdot 9 + 5 \cdot 9) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.5

Multiply $9$ by $- 1$ .

$\frac{(4 x - 12 - 9 y + 5 \cdot 9) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.6

Multiply $5$ by $9$ .

$\frac{(4 x - 12 - 9 y + 45) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.7

Add $- 12$ and $45$ .

$\frac{(4 x - 9 y + 33) ((x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.8

Apply the distributive property.

$\frac{(4 x - 9 y + 33) (x \cdot 4 - 3 \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.9

Move $4$ to the left of $x$ .

$\frac{(4 x - 9 y + 33) (4 \cdot x - 3 \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.10

Multiply $- 3$ by $4$ .

$\frac{(4 x - 9 y + 33) (4 \cdot x - 12 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 4.5.4.11

Apply the distributive property.

$\frac{(4 x - 9 y + 33) (4 x - 12 - (- y \cdot 9 + 5 \cdot 9))}{1296} = 1$

Step 4.5.4.12

Multiply $9$ by $- 1$ .

$\frac{(4 x - 9 y + 33) (4 x - 12 - (- 9 y + 5 \cdot 9))}{1296} = 1$

Step 4.5.4.13

Multiply $5$ by $9$ .

$\frac{(4 x - 9 y + 33) (4 x - 12 - (- 9 y + 45))}{1296} = 1$

Step 4.5.4.14

Apply the distributive property.

$\frac{(4 x - 9 y + 33) (4 x - 12 - (- 9 y) - 1 \cdot 45)}{1296} = 1$

Step 4.5.4.15

Multiply $- 9$ by $- 1$ .

$\frac{(4 x - 9 y + 33) (4 x - 12 + 9 y - 1 \cdot 45)}{1296} = 1$

Step 4.5.4.16

Multiply $- 1$ by $45$ .

$\frac{(4 x - 9 y + 33) (4 x - 12 + 9 y - 45)}{1296} = 1$

Step 4.5.4.17

Subtract $45$ from $- 12$ .

$\frac{(4 x - 9 y + 33) (4 x + 9 y - 57)}{1296} = 1$

Step 5

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 6

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

$\frac{{(- x - 3)}^{2}}{81} - \frac{{(y + 5)}^{2}}{16} = 1$

Step 7

Simplify.

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Step 7.1

To write $\frac{{(- x - 3)}^{2}}{81}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{{(- x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{{(y + 5)}^{2}}{16} = 1$

Step 7.2

To write $- \frac{{(y + 5)}^{2}}{16}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$\frac{{(- x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{{(y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 7.3

Write each expression with a common denominator of $1296$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.1

Multiply $\frac{{(- x - 3)}^{2}}{81}$ by $\frac{16}{16}$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{81 \cdot 16} - \frac{{(y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 7.3.2

Multiply $81$ by $16$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{1296} - \frac{{(y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 7.3.3

Multiply $\frac{{(y + 5)}^{2}}{16}$ by $\frac{81}{81}$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{1296} - \frac{{(y + 5)}^{2} \cdot 81}{16 \cdot 81} = 1$

Step 7.3.4

Multiply $16$ by $81$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{1296} - \frac{{(y + 5)}^{2} \cdot 81}{1296} = 1$

Step 7.4

Combine the numerators over the common denominator.

$\frac{{(- x - 3)}^{2} \cdot 16 - {(y + 5)}^{2} \cdot 81}{1296} = 1$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Rewrite ${(- x - 3)}^{2} \cdot 16$ as ${((- x - 3) \cdot 4)}^{2}$ .

$\frac{{((- x - 3) \cdot 4)}^{2} - {(y + 5)}^{2} \cdot 81}{1296} = 1$

Step 7.5.2

Rewrite ${(y + 5)}^{2} \cdot 81$ as ${((y + 5) \cdot 9)}^{2}$ .

$\frac{{((- x - 3) \cdot 4)}^{2} - {((y + 5) \cdot 9)}^{2}}{1296} = 1$

Step 7.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = (- x - 3) \cdot 4$ and $b = (y + 5) \cdot 9$ .

$\frac{((- x - 3) \cdot 4 + (y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4

Simplify.

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Step 7.5.4.1

Apply the distributive property.

$\frac{(- x \cdot 4 - 3 \cdot 4 + (y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.2

Multiply $4$ by $- 1$ .

$\frac{(- 4 x - 3 \cdot 4 + (y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.3

Multiply $- 3$ by $4$ .

$\frac{(- 4 x - 12 + (y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.4

Apply the distributive property.

$\frac{(- 4 x - 12 + y \cdot 9 + 5 \cdot 9) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.5

Move $9$ to the left of $y$ .

$\frac{(- 4 x - 12 + 9 \cdot y + 5 \cdot 9) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.6

Multiply $5$ by $9$ .

$\frac{(- 4 x - 12 + 9 \cdot y + 45) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.7

Add $- 12$ and $45$ .

$\frac{(- 4 x + 9 y + 33) ((- x - 3) \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.8

Apply the distributive property.

$\frac{(- 4 x + 9 y + 33) (- x \cdot 4 - 3 \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.9

Multiply $4$ by $- 1$ .

$\frac{(- 4 x + 9 y + 33) (- 4 x - 3 \cdot 4 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.10

Multiply $- 3$ by $4$ .

$\frac{(- 4 x + 9 y + 33) (- 4 x - 12 - ((y + 5) \cdot 9))}{1296} = 1$

Step 7.5.4.11

Apply the distributive property.

$\frac{(- 4 x + 9 y + 33) (- 4 x - 12 - (y \cdot 9 + 5 \cdot 9))}{1296} = 1$

Step 7.5.4.12

Move $9$ to the left of $y$ .

$\frac{(- 4 x + 9 y + 33) (- 4 x - 12 - (9 \cdot y + 5 \cdot 9))}{1296} = 1$

Step 7.5.4.13

Multiply $5$ by $9$ .

$\frac{(- 4 x + 9 y + 33) (- 4 x - 12 - (9 \cdot y + 45))}{1296} = 1$

Step 7.5.4.14

Apply the distributive property.

$\frac{(- 4 x + 9 y + 33) (- 4 x - 12 - (9 y) - 1 \cdot 45)}{1296} = 1$

Step 7.5.4.15

Multiply $9$ by $- 1$ .

$\frac{(- 4 x + 9 y + 33) (- 4 x - 12 - 9 y - 1 \cdot 45)}{1296} = 1$

Step 7.5.4.16

Multiply $- 1$ by $45$ .

$\frac{(- 4 x + 9 y + 33) (- 4 x - 12 - 9 y - 45)}{1296} = 1$

Step 7.5.4.17

Subtract $45$ from $- 12$ .

$\frac{(- 4 x + 9 y + 33) (- 4 x - 9 y - 57)}{1296} = 1$

Step 8

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 9

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

$\frac{{(- x - 3)}^{2}}{81} - \frac{{(- y + 5)}^{2}}{16} = 1$

Step 10

Simplify.

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Step 10.1

To write $\frac{{(- x - 3)}^{2}}{81}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{{(- x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{{(- y + 5)}^{2}}{16} = 1$

Step 10.2

To write $- \frac{{(- y + 5)}^{2}}{16}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$\frac{{(- x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{{(- y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 10.3

Write each expression with a common denominator of $1296$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.3.1

Multiply $\frac{{(- x - 3)}^{2}}{81}$ by $\frac{16}{16}$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{81 \cdot 16} - \frac{{(- y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 10.3.2

Multiply $81$ by $16$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{1296} - \frac{{(- y + 5)}^{2}}{16} \cdot \frac{81}{81} = 1$

Step 10.3.3

Multiply $\frac{{(- y + 5)}^{2}}{16}$ by $\frac{81}{81}$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{1296} - \frac{{(- y + 5)}^{2} \cdot 81}{16 \cdot 81} = 1$

Step 10.3.4

Multiply $16$ by $81$ .

$\frac{{(- x - 3)}^{2} \cdot 16}{1296} - \frac{{(- y + 5)}^{2} \cdot 81}{1296} = 1$

Step 10.4

Combine the numerators over the common denominator.

$\frac{{(- x - 3)}^{2} \cdot 16 - {(- y + 5)}^{2} \cdot 81}{1296} = 1$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Rewrite ${(- x - 3)}^{2} \cdot 16$ as ${((- x - 3) \cdot 4)}^{2}$ .

$\frac{{((- x - 3) \cdot 4)}^{2} - {(- y + 5)}^{2} \cdot 81}{1296} = 1$

Step 10.5.2

Rewrite ${(- y + 5)}^{2} \cdot 81$ as ${((- y + 5) \cdot 9)}^{2}$ .

$\frac{{((- x - 3) \cdot 4)}^{2} - {((- y + 5) \cdot 9)}^{2}}{1296} = 1$

Step 10.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = (- x - 3) \cdot 4$ and $b = (- y + 5) \cdot 9$ .

$\frac{((- x - 3) \cdot 4 + (- y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4

Simplify.

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Step 10.5.4.1

Apply the distributive property.

$\frac{(- x \cdot 4 - 3 \cdot 4 + (- y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.2

Multiply $4$ by $- 1$ .

$\frac{(- 4 x - 3 \cdot 4 + (- y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.3

Multiply $- 3$ by $4$ .

$\frac{(- 4 x - 12 + (- y + 5) \cdot 9) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.4

Apply the distributive property.

$\frac{(- 4 x - 12 - y \cdot 9 + 5 \cdot 9) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.5

Multiply $9$ by $- 1$ .

$\frac{(- 4 x - 12 - 9 y + 5 \cdot 9) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.6

Multiply $5$ by $9$ .

$\frac{(- 4 x - 12 - 9 y + 45) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.7

Add $- 12$ and $45$ .

$\frac{(- 4 x - 9 y + 33) ((- x - 3) \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.8

Apply the distributive property.

$\frac{(- 4 x - 9 y + 33) (- x \cdot 4 - 3 \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.9

Multiply $4$ by $- 1$ .

$\frac{(- 4 x - 9 y + 33) (- 4 x - 3 \cdot 4 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.10

Multiply $- 3$ by $4$ .

$\frac{(- 4 x - 9 y + 33) (- 4 x - 12 - ((- y + 5) \cdot 9))}{1296} = 1$

Step 10.5.4.11

Apply the distributive property.

$\frac{(- 4 x - 9 y + 33) (- 4 x - 12 - (- y \cdot 9 + 5 \cdot 9))}{1296} = 1$

Step 10.5.4.12

Multiply $9$ by $- 1$ .

$\frac{(- 4 x - 9 y + 33) (- 4 x - 12 - (- 9 y + 5 \cdot 9))}{1296} = 1$

Step 10.5.4.13

Multiply $5$ by $9$ .

$\frac{(- 4 x - 9 y + 33) (- 4 x - 12 - (- 9 y + 45))}{1296} = 1$

Step 10.5.4.14

Apply the distributive property.

$\frac{(- 4 x - 9 y + 33) (- 4 x - 12 - (- 9 y) - 1 \cdot 45)}{1296} = 1$

Step 10.5.4.15

Multiply $- 9$ by $- 1$ .

$\frac{(- 4 x - 9 y + 33) (- 4 x - 12 + 9 y - 1 \cdot 45)}{1296} = 1$

Step 10.5.4.16

Multiply $- 1$ by $45$ .

$\frac{(- 4 x - 9 y + 33) (- 4 x - 12 + 9 y - 45)}{1296} = 1$

Step 10.5.4.17

Subtract $45$ from $- 12$ .

$\frac{(- 4 x - 9 y + 33) (- 4 x + 9 y - 57)}{1296} = 1$

Step 11

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

Not symmetric to the origin

Step 12

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 13