Algebra Examples

$x^{3} + y^{3} = 126$ $x^{2} - x y + y^{2} = 21$

Step 1

Solve for $x$ in $x^{3} + y^{3} = 126$ .

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Step 1.1

Subtract $y^{3}$ from both sides of the equation.

$x^{3} = 126 - y^{3}$

$x^{2} - x y + y^{2} = 21$

Step 1.2

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

$x = \sqrt[3]{126 - y^{3}}$

$x^{2} - x y + y^{2} = 21$

$x = \sqrt[3]{126 - y^{3}}$

$x^{2} - x y + y^{2} = 21$

Step 2

Replace all occurrences of $x$ with $\sqrt[3]{126 - y^{3}}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} - x y + y^{2} = 21$ with $\sqrt[3]{126 - y^{3}}$ .

${(\sqrt[3]{126 - y^{3}})}^{2} - \sqrt[3]{126 - y^{3}} \cdot y + y^{2} = 21$

$x = \sqrt[3]{126 - y^{3}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Rewrite ${\sqrt[3]{126 - y^{3}}}^{2}$ as $\sqrt[3]{{(126 - y^{3})}^{2}}$ .

$\sqrt[3]{{(126 - y^{3})}^{2}} - \sqrt[3]{126 - y^{3}} y + y^{2} = 21$

$x = \sqrt[3]{126 - y^{3}}$

$\sqrt[3]{{(126 - y^{3})}^{2}} - \sqrt[3]{126 - y^{3}} y + y^{2} = 21$

$x = \sqrt[3]{126 - y^{3}}$

$\sqrt[3]{{(126 - y^{3})}^{2}} - \sqrt[3]{126 - y^{3}} y + y^{2} = 21$

$x = \sqrt[3]{126 - y^{3}}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$y = 1, 5$

$x = \sqrt[3]{126 - y^{3}}$

Step 4

Replace all occurrences of $y$ with $1$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \sqrt[3]{126 - y^{3}}$ with $1$ .

$x = \sqrt[3]{126 - {(1)}^{3}}$

$y = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\sqrt[3]{126 - {(1)}^{3}}$ .

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Step 4.2.1.1

One to any power is one.

$x = \sqrt[3]{126 - 1 \cdot 1}$

$y = 1$

Step 4.2.1.2

Multiply $- 1$ by $1$ .

$x = \sqrt[3]{126 - 1}$

$y = 1$

Step 4.2.1.3

Subtract $1$ from $126$ .

$x = \sqrt[3]{125}$

$y = 1$

Step 4.2.1.4

Rewrite $125$ as $5^{3}$ .

$x = \sqrt[3]{5^{3}}$

$y = 1$

Step 4.2.1.5

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

$x = 5$

$y = 1$

$x = 5$

$y = 1$

$x = 5$

$y = 1$

$x = 5$

$y = 1$

Step 5

Replace all occurrences of $y$ with $5$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \sqrt[3]{126 - y^{3}}$ with $5$ .

$x = \sqrt[3]{126 - {(5)}^{3}}$

$y = 5$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\sqrt[3]{126 - {(5)}^{3}}$ .

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Step 5.2.1.1

Raise $5$ to the power of $3$ .

$x = \sqrt[3]{126 - 1 \cdot 125}$

$y = 5$

Step 5.2.1.2

Multiply $- 1$ by $125$ .

$x = \sqrt[3]{126 - 125}$

$y = 5$

Step 5.2.1.3

Subtract $125$ from $126$ .

$x = \sqrt[3]{1}$

$y = 5$

Step 5.2.1.4

Any root of $1$ is $1$ .

$x = 1$

$y = 5$

$x = 1$

$y = 5$

$x = 1$

$y = 5$

$x = 1$

$y = 5$

Step 6

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

$(5, 1)$

$(1, 5)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(5, 1), (1, 5)$

Equation Form:

$x = 5, y = 1$

$x = 1, y = 5$

Step 8