Precalculus Examples

$2 x^{2} - 27 y^{3} = 19$ , $4 x^{2} + 54 y^{3} = 34$

Step 1

Solve for $y$ in $2 x^{2} - 27 y^{3} = 19$ .

Tap for more steps...

Step 1.1

Subtract $2 x^{2}$ from both sides of the equation.

$- 27 y^{3} = 19 - 2 x^{2}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.2

Divide each term in $- 27 y^{3} = 19 - 2 x^{2}$ by $- 27$ and simplify.

Tap for more steps...

Step 1.2.1

Divide each term in $- 27 y^{3} = 19 - 2 x^{2}$ by $- 27$ .

$\frac{- 27 y^{3}}{- 27} = \frac{19}{- 27} + \frac{- 2 x^{2}}{- 27}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.1

Cancel the common factor of $- 27$ .

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 27 y^{3}}{- 27} = \frac{19}{- 27} + \frac{- 2 x^{2}}{- 27}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.2.2.1.2

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

$y^{3} = \frac{19}{- 27} + \frac{- 2 x^{2}}{- 27}$

$4 x^{2} + 54 y^{3} = 34$

$y^{3} = \frac{19}{- 27} + \frac{- 2 x^{2}}{- 27}$

$4 x^{2} + 54 y^{3} = 34$

$y^{3} = \frac{19}{- 27} + \frac{- 2 x^{2}}{- 27}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.2.3.1.1

Move the negative in front of the fraction.

$y^{3} = - \frac{19}{27} + \frac{- 2 x^{2}}{- 27}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.2.3.1.2

Dividing two negative values results in a positive value.

$y^{3} = - \frac{19}{27} + \frac{2 x^{2}}{27}$

$4 x^{2} + 54 y^{3} = 34$

$y^{3} = - \frac{19}{27} + \frac{2 x^{2}}{27}$

$4 x^{2} + 54 y^{3} = 34$

$y^{3} = - \frac{19}{27} + \frac{2 x^{2}}{27}$

$4 x^{2} + 54 y^{3} = 34$

$y^{3} = - \frac{19}{27} + \frac{2 x^{2}}{27}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.3

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

$y = \sqrt[3]{- \frac{19}{27} + \frac{2 x^{2}}{27}}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.4

Simplify $\sqrt[3]{- \frac{19}{27} + \frac{2 x^{2}}{27}}$ .

Tap for more steps...

Step 1.4.1

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{- 19 + 2 x^{2}}{27}}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.4.2

Reorder $- 19$ and $2 x^{2}$ .

$y = \sqrt[3]{\frac{2 x^{2} - 19}{27}}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.4.3

Rewrite $\frac{2 x^{2} - 19}{27}$ as ${(\frac{1}{3})}^{3} (2 x^{2} - 19)$ .

Tap for more steps...

Step 1.4.3.1

Factor the perfect power $1^{3}$ out of $2 x^{2} - 19$ .

$y = \sqrt[3]{\frac{1^{3} (2 x^{2} - 19)}{27}}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.4.3.2

Factor the perfect power $3^{3}$ out of $27$ .

$y = \sqrt[3]{\frac{1^{3} (2 x^{2} - 19)}{3^{3} \cdot 1}}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.4.3.3

Rearrange the fraction $\frac{1^{3} (2 x^{2} - 19)}{3^{3} \cdot 1}$ .

$y = \sqrt[3]{{(\frac{1}{3})}^{3} (2 x^{2} - 19)}$

$4 x^{2} + 54 y^{3} = 34$

$y = \sqrt[3]{{(\frac{1}{3})}^{3} (2 x^{2} - 19)}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.4.4

Pull terms out from under the radical.

$y = \frac{1}{3} \cdot \sqrt[3]{2 x^{2} - 19}$

$4 x^{2} + 54 y^{3} = 34$

Step 1.4.5

Combine $\frac{1}{3}$ and $\sqrt[3]{2 x^{2} - 19}$ .

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$4 x^{2} + 54 y^{3} = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$4 x^{2} + 54 y^{3} = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$4 x^{2} + 54 y^{3} = 34$

Step 2

Replace all occurrences of $y$ with $\frac{\sqrt[3]{2 x^{2} - 19}}{3}$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $y$ in $4 x^{2} + 54 y^{3} = 34$ with $\frac{\sqrt[3]{2 x^{2} - 19}}{3}$ .

$4 x^{2} + 54 {(\frac{\sqrt[3]{2 x^{2} - 19}}{3})}^{3} = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify $4 x^{2} + 54 {(\frac{\sqrt[3]{2 x^{2} - 19}}{3})}^{3}$ .

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Apply the product rule to $\frac{\sqrt[3]{2 x^{2} - 19}}{3}$ .

$4 x^{2} + 54 (\frac{{\sqrt[3]{2 x^{2} - 19}}^{3}}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.2

Rewrite ${\sqrt[3]{2 x^{2} - 19}}^{3}$ as $2 x^{2} - 19$ .

Tap for more steps...

Step 2.2.1.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x^{2} - 19}$ as ${(2 x^{2} - 19)}^{\frac{1}{3}}$ .

$4 x^{2} + 54 (\frac{{({(2 x^{2} - 19)}^{\frac{1}{3}})}^{3}}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 x^{2} + 54 (\frac{{(2 x^{2} - 19)}^{\frac{1}{3} \cdot 3}}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.2.3

Combine $\frac{1}{3}$ and $3$ .

$4 x^{2} + 54 (\frac{{(2 x^{2} - 19)}^{\frac{3}{3}}}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.2.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1.2.4.1

Cancel the common factor.

$4 x^{2} + 54 (\frac{{(2 x^{2} - 19)}^{\frac{3}{3}}}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.2.4.2

Rewrite the expression.

$4 x^{2} + 54 (\frac{2 x^{2} - 19}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$4 x^{2} + 54 (\frac{2 x^{2} - 19}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.2.5

Simplify.

$4 x^{2} + 54 (\frac{2 x^{2} - 19}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$4 x^{2} + 54 (\frac{2 x^{2} - 19}{3^{3}}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.3

Raise $3$ to the power of $3$ .

$4 x^{2} + 54 (\frac{2 x^{2} - 19}{27}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.4

Cancel the common factor of $27$ .

Tap for more steps...

Step 2.2.1.1.4.1

Factor $27$ out of $54$ .

$4 x^{2} + 27 (2) (\frac{2 x^{2} - 19}{27}) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.4.2

Cancel the common factor.

$4 x^{2} + 27 \cdot (2 (\frac{2 x^{2} - 19}{27})) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.4.3

Rewrite the expression.

$4 x^{2} + 2 (2 x^{2} - 19) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$4 x^{2} + 2 (2 x^{2} - 19) = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.5

Apply the distributive property.

$4 x^{2} + 2 (2 x^{2}) + 2 \cdot - 19 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.6

Multiply $2$ by $2$ .

$4 x^{2} + 4 x^{2} + 2 \cdot - 19 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.1.7

Multiply $2$ by $- 19$ .

$4 x^{2} + 4 x^{2} - 38 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$4 x^{2} + 4 x^{2} - 38 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 2.2.1.2

Add $4 x^{2}$ and $4 x^{2}$ .

$8 x^{2} - 38 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$8 x^{2} - 38 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$8 x^{2} - 38 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$8 x^{2} - 38 = 34$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3

Solve for $x$ in $8 x^{2} - 38 = 34$ .

Tap for more steps...

Step 3.1

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 3.1.1

Add $38$ to both sides of the equation.

$8 x^{2} = 34 + 38$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.1.2

Add $34$ and $38$ .

$8 x^{2} = 72$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$8 x^{2} = 72$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.2

Divide each term in $8 x^{2} = 72$ by $8$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $8 x^{2} = 72$ by $8$ .

$\frac{8 x^{2}}{8} = \frac{72}{8}$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{8 x^{2}}{8} = \frac{72}{8}$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{72}{8}$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$x^{2} = \frac{72}{8}$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$x^{2} = \frac{72}{8}$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Divide $72$ by $8$ .

$x^{2} = 9$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$x^{2} = 9$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$x^{2} = 9$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.3

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

$x = \pm \sqrt{9}$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.4

Simplify $\pm \sqrt{9}$ .

Tap for more steps...

Step 3.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.4.2

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

$x = \pm 3$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$x = \pm 3$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$x = 3$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.5.2

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

$x = - 3$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 3.5.3

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

$x = 3, - 3$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$x = 3, - 3$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

$x = 3, - 3$

$y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$

Step 4

Replace all occurrences of $x$ with $3$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $x$ in $y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$ with $3$ .

$y = \frac{\sqrt[3]{2 {(3)}^{2} - 19}}{3}$

$x = 3$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $\frac{\sqrt[3]{2 {(3)}^{2} - 19}}{3}$ .

Tap for more steps...

Step 4.2.1.1

Simplify the numerator.

Tap for more steps...

Step 4.2.1.1.1

Raise $3$ to the power of $2$ .

$y = \frac{\sqrt[3]{2 \cdot 9 - 19}}{3}$

$x = 3$

Step 4.2.1.1.2

Multiply $2$ by $9$ .

$y = \frac{\sqrt[3]{18 - 19}}{3}$

$x = 3$

Step 4.2.1.1.3

Subtract $19$ from $18$ .

$y = \frac{\sqrt[3]{- 1}}{3}$

$x = 3$

Step 4.2.1.1.4

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \frac{\sqrt[3]{{(- 1)}^{3}}}{3}$

$x = 3$

Step 4.2.1.1.5

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

$y = \frac{- 1}{3}$

$x = 3$

$y = \frac{- 1}{3}$

$x = 3$

Step 4.2.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{3}$

$x = 3$

$y = - \frac{1}{3}$

$x = 3$

$y = - \frac{1}{3}$

$x = 3$

$y = - \frac{1}{3}$

$x = 3$

Step 5

Replace all occurrences of $x$ with $- 3$ in each equation.

Tap for more steps...

Step 5.1

Replace all occurrences of $x$ in $y = \frac{\sqrt[3]{2 x^{2} - 19}}{3}$ with $- 3$ .

$y = \frac{\sqrt[3]{2 {(- 3)}^{2} - 19}}{3}$

$x = - 3$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $\frac{\sqrt[3]{2 {(- 3)}^{2} - 19}}{3}$ .

Tap for more steps...

Step 5.2.1.1

Simplify the numerator.

Tap for more steps...

Step 5.2.1.1.1

Raise $- 3$ to the power of $2$ .

$y = \frac{\sqrt[3]{2 \cdot 9 - 19}}{3}$

$x = - 3$

Step 5.2.1.1.2

Multiply $2$ by $9$ .

$y = \frac{\sqrt[3]{18 - 19}}{3}$

$x = - 3$

Step 5.2.1.1.3

Subtract $19$ from $18$ .

$y = \frac{\sqrt[3]{- 1}}{3}$

$x = - 3$

Step 5.2.1.1.4

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \frac{\sqrt[3]{{(- 1)}^{3}}}{3}$

$x = - 3$

Step 5.2.1.1.5

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

$y = \frac{- 1}{3}$

$x = - 3$

$y = \frac{- 1}{3}$

$x = - 3$

Step 5.2.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{3}$

$x = - 3$

$y = - \frac{1}{3}$

$x = - 3$

$y = - \frac{1}{3}$

$x = - 3$

$y = - \frac{1}{3}$

$x = - 3$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(3, - \frac{1}{3})$

$(- 3, - \frac{1}{3})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(3, - \frac{1}{3}), (- 3, - \frac{1}{3})$

Equation Form:

$x = 3, y = - \frac{1}{3}$

$x = - 3, y = - \frac{1}{3}$

Step 8