Precalculus Examples

$2 x^{3} + 2 y^{3} = 9 x y$ , $(2, 1)$

Step 1

Insert the values of $x = 2$ and $y = 1$ into the equation to find if the ordered pair is a solution.

$2 {(2)}^{3} + 2 {(1)}^{3} = 9 (2) \cdot (1)$

Step 2

Simplify $2 \cdot 2^{3} + 2 \cdot 1^{3}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2$ by $2^{3}$ by adding the exponents.

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Step 2.1.1.1

Multiply $2$ by $2^{3}$ .

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Step 2.1.1.1.1

Raise $2$ to the power of $1$ .

$2^{1} \cdot 2^{3} + 2 \cdot 1^{3} = 9 (2) \cdot 1$

Step 2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 3} + 2 \cdot 1^{3} = 9 (2) \cdot 1$

Step 2.1.1.2

Add $1$ and $3$ .

$2^{4} + 2 \cdot 1^{3} = 9 (2) \cdot 1$

Step 2.1.2

Raise $2$ to the power of $4$ .

$16 + 2 \cdot 1^{3} = 9 (2) \cdot 1$

Step 2.1.3

One to any power is one.

$16 + 2 \cdot 1 = 9 (2) \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$16 + 2 = 9 (2) \cdot 1$

Step 2.2

Add $16$ and $2$ .

$18 = 9 (2) \cdot 1$

Step 3

Multiply $9 (2) \cdot 1$ .

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Step 3.1

Multiply $9$ by $2$ .

$18 = 18 \cdot 1$

Step 3.2

Multiply $18$ by $1$ .

$18 = 18$

Step 4

Since $18 = 18$ , the equation will always be true.

Always true

Step 5

Since the equation is always true when the values are used, the ordered pair is a solution.

The ordered pair is a solution to the equation.