Precalculus Examples

$x^{4} + y^{3} = 264$ , $3 x^{4} + 5 y^{3} = 808$

Step 1

Simplify the left side.

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Step 1.1

Reorder $x^{4}$ and $y^{3}$ .

$y^{3} + x^{4} = 264$

$3 x^{4} + 5 y^{3} = 808$

$y^{3} + x^{4} = 264$

$3 x^{4} + 5 y^{3} = 808$

Step 2

Simplify the left side.

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Step 2.1

Reorder $3 x^{4}$ and $5 y^{3}$ .

$5 y^{3} + 3 x^{4} = 808$

$y^{3} + x^{4} = 264$

$5 y^{3} + 3 x^{4} = 808$

$y^{3} + x^{4} = 264$

Step 3

Multiply each equation by the value that makes the coefficients of $x^{4}$ opposite.

$- 3 \cdot (y^{3} + x^{4}) = (- 3) (264)$

$5 y^{3} + 3 x^{4} = 808$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Apply the distributive property.

$- 3 y^{3} - 3 x^{4} = (- 3) (264)$

$5 y^{3} + 3 x^{4} = 808$

$- 3 y^{3} - 3 x^{4} = (- 3) (264)$

$5 y^{3} + 3 x^{4} = 808$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 3$ by $264$ .

$- 3 y^{3} - 3 x^{4} = - 792$

$5 y^{3} + 3 x^{4} = 808$

$- 3 y^{3} - 3 x^{4} = - 792$

$5 y^{3} + 3 x^{4} = 808$

$- 3 y^{3} - 3 x^{4} = - 792$

$5 y^{3} + 3 x^{4} = 808$

Step 5

Add the two equations together to eliminate $x^{4}$ from the system.

	$-$	$3$	$y^{3}$	$-$	$3$	$x^{4}$	$=$	$-$	$7$	$9$	$2$
$+$		$5$	$y^{3}$	$+$	$3$	$x^{4}$	$=$		$8$	$0$	$8$
		$2$	$y^{3}$				$=$			$1$	$6$

Step 6

Divide each term in $2 y^{3} = 16$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 y^{3} = 16$ by $2$ .

$\frac{2 y^{3}}{2} = \frac{16}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 y^{3}}{2} = \frac{16}{2}$

Step 6.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{16}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $16$ by $2$ .

$y^{3} = 8$

Step 7

Substitute the value found for $y^{3}$ into one of the original equations, then solve for $x^{4}$ .

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Step 7.1

Substitute the value found for $y^{3}$ into one of the original equations to solve for $x^{4}$ .

$- 3 \cdot 8 - 3 x^{4} = - 792$

Step 7.2

Multiply $- 3$ by $8$ .

$- 24 - 3 x^{4} = - 792$

Step 7.3

Move all terms not containing $x^{4}$ to the right side of the equation.

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Step 7.3.1

Add $24$ to both sides of the equation.

$- 3 x^{4} = - 792 + 24$

Step 7.3.2

Add $- 792$ and $24$ .

$- 3 x^{4} = - 768$

Step 7.4

Divide each term in $- 3 x^{4} = - 768$ by $- 3$ and simplify.

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Step 7.4.1

Divide each term in $- 3 x^{4} = - 768$ by $- 3$ .

$\frac{- 3 x^{4}}{- 3} = \frac{- 768}{- 3}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $- 3$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{- 3 x^{4}}{- 3} = \frac{- 768}{- 3}$

Step 7.4.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{- 768}{- 3}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Divide $- 768$ by $- 3$ .

$x^{4} = 256$

Step 8

This is the final solution to the independent system of equations.

$y^{3} = 8$

$x^{4} = 256$

Step 9

Subtract $8$ from both sides of the equation.

$y^{3} - 8 = 0$

$x^{4} = 256$

Step 10

Factor the left side of the equation.

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Step 10.1

Rewrite $8$ as $2^{3}$ .

$y^{3} - 2^{3} = 0$

$x^{4} = 256$

Step 10.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 2$ .

$(y - 2) (y^{2} + y \cdot 2 + 2^{2}) = 0$

$x^{4} = 256$

Step 10.3

Simplify.

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Step 10.3.1

Move $2$ to the left of $y$ .

$(y - 2) (y^{2} + 2 \cdot y + 2^{2}) = 0$

$x^{4} = 256$

Step 10.3.2

Multiply $2$ by $y$ .

$(y - 2) (y^{2} + 2 y + 2^{2}) = 0$

$x^{4} = 256$

Step 10.3.3

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

$(y - 2) (y^{2} + 2 y + 4) = 0$

$x^{4} = 256$

$(y - 2) (y^{2} + 2 y + 4) = 0$

$x^{4} = 256$

$(y - 2) (y^{2} + 2 y + 4) = 0$

$x^{4} = 256$

Step 11

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 2 = 0$

$y^{2} + 2 y + 4 = 0$

$x^{4} = 256$

Step 12

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 12.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

$x^{4} = 256$

Step 12.2

Add $2$ to both sides of the equation.

$y = 2$

$x^{4} = 256$

$y = 2$

$x^{4} = 256$

Step 13

Set $y^{2} + 2 y + 4$ equal to $0$ and solve for $y$ .

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Step 13.1

Set $y^{2} + 2 y + 4$ equal to $0$ .

$y^{2} + 2 y + 4 = 0$

$x^{4} = 256$

Step 13.2

Solve $y^{2} + 2 y + 4 = 0$ for $y$ .

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Step 13.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x^{4} = 256$

Step 13.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3

Simplify.

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Step 13.2.3.1

Simplify the numerator.

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Step 13.2.3.1.1

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 13.2.3.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.3

Subtract $16$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$y = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 13.2.3.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.7.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.1.9

Move $2$ to the left of $i$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.3.2

Multiply $2$ by $1$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

$x^{4} = 256$

Step 13.2.3.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$y = - 1 \pm i \sqrt{3}$

$x^{4} = 256$

$y = - 1 \pm i \sqrt{3}$

$x^{4} = 256$

Step 13.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 13.2.4.1

Simplify the numerator.

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Step 13.2.4.1.1

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 13.2.4.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.3

Subtract $16$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$y = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 13.2.4.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.1.9

Move $2$ to the left of $i$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.4.2

Multiply $2$ by $1$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

$x^{4} = 256$

Step 13.2.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$y = - 1 \pm i \sqrt{3}$

$x^{4} = 256$

Step 13.2.4.4

Change the $\pm$ to $+$ .

$y = - 1 + i \sqrt{3}$

$x^{4} = 256$

$y = - 1 + i \sqrt{3}$

$x^{4} = 256$

Step 13.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 13.2.5.1

Simplify the numerator.

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Step 13.2.5.1.1

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

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 13.2.5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.3

Subtract $16$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$y = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$y = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 13.2.5.1.7.1

Factor $4$ out of $12$ .

$y = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.1.9

Move $2$ to the left of $i$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x^{4} = 256$

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x^{4} = 256$

Step 13.2.5.2

Multiply $2$ by $1$ .

$y = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

$x^{4} = 256$

Step 13.2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$y = - 1 \pm i \sqrt{3}$

$x^{4} = 256$

Step 13.2.5.4

Change the $\pm$ to $-$ .

$y = - 1 - i \sqrt{3}$

$x^{4} = 256$

$y = - 1 - i \sqrt{3}$

$x^{4} = 256$

Step 13.2.6

The final answer is the combination of both solutions.

$y = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$x^{4} = 256$

$y = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$x^{4} = 256$

$y = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$x^{4} = 256$

Step 14

The final solution is all the values that make $(y - 2) (y^{2} + 2 y + 4) = 0$ true.

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$x^{4} = 256$

Step 15

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{256}$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 16

Simplify $\pm \sqrt[4]{256}$ .

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Step 16.1

Rewrite $256$ as $4^{4}$ .

$x = \pm \sqrt[4]{4^{4}}$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 16.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 4$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$x = \pm 4$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 17

The complete solution is the result of both the positive and negative portions of the solution.

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Step 17.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 17.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 17.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

$x = 4, - 4$

$y = 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 18

The final result is the combination of all values of $x$ with all values of $y$ .

$(4, 2), (4, - 1 + i \sqrt{3}), (4, - 1 - i \sqrt{3}), (- 4, 2), (- 4, - 1 + i \sqrt{3}), (- 4, - 1 - i \sqrt{3})$

Step 19