Precalculus Examples

$3 x + y = 2$ , $x^{3} - 2 + y = 0$

Step 1

Subtract $3 x$ from both sides of the equation.

$y = 2 - 3 x$

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

Step 2

Replace all occurrences of $y$ with $2 - 3 x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{3} - 2 + y = 0$ with $2 - 3 x$ .

$x^{3} - 2 + 2 - 3 x = 0$

$y = 2 - 3 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{3} - 2 + 2 - 3 x$ .

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Step 2.2.1.1

Remove parentheses.

$x^{3} - 2 + 2 - 3 x = 0$

$y = 2 - 3 x$

Step 2.2.1.2

Combine the opposite terms in $x^{3} - 2 + 2 - 3 x$ .

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Step 2.2.1.2.1

Add $- 2$ and $2$ .

$x^{3} + 0 - 3 x = 0$

$y = 2 - 3 x$

Step 2.2.1.2.2

Add $x^{3}$ and $0$ .

$x^{3} - 3 x = 0$

$y = 2 - 3 x$

$x^{3} - 3 x = 0$

$y = 2 - 3 x$

$x^{3} - 3 x = 0$

$y = 2 - 3 x$

$x^{3} - 3 x = 0$

$y = 2 - 3 x$

$x^{3} - 3 x = 0$

$y = 2 - 3 x$

Step 3

Solve for $x$ in $x^{3} - 3 x = 0$ .

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Step 3.1

Factor $x$ out of $x^{3} - 3 x$ .

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Step 3.1.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 3 x = 0$

$y = 2 - 3 x$

Step 3.1.2

Factor $x$ out of $- 3 x$ .

$x \cdot x^{2} + x \cdot - 3 = 0$

$y = 2 - 3 x$

Step 3.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 3$ .

$x (x^{2} - 3) = 0$

$y = 2 - 3 x$

$x (x^{2} - 3) = 0$

$y = 2 - 3 x$

Step 3.2

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

$x = 0$

$x^{2} - 3 = 0$

$y = 2 - 3 x$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

$y = 2 - 3 x$

Step 3.4

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

$y = 2 - 3 x$

Step 3.4.2

Solve $x^{2} - 3 = 0$ for $x$ .

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Step 3.4.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

$y = 2 - 3 x$

Step 3.4.2.2

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

$x = \pm \sqrt{3}$

$y = 2 - 3 x$

Step 3.4.2.3

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

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Step 3.4.2.3.1

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

$x = \sqrt{3}$

$y = 2 - 3 x$

Step 3.4.2.3.2

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

$x = - \sqrt{3}$

$y = 2 - 3 x$

Step 3.4.2.3.3

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

$x = \sqrt{3}, - \sqrt{3}$

$y = 2 - 3 x$

$x = \sqrt{3}, - \sqrt{3}$

$y = 2 - 3 x$

$x = \sqrt{3}, - \sqrt{3}$

$y = 2 - 3 x$

$x = \sqrt{3}, - \sqrt{3}$

$y = 2 - 3 x$

Step 3.5

The final solution is all the values that make $x (x^{2} - 3) = 0$ true.

$x = 0, \sqrt{3}, - \sqrt{3}$

$y = 2 - 3 x$

$x = 0, \sqrt{3}, - \sqrt{3}$

$y = 2 - 3 x$

Step 4

Replace all occurrences of $x$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 2 - 3 x$ with $0$ .

$y = 2 - 3 \cdot 0$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 - 3 \cdot 0$ .

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Step 4.2.1.1

Multiply $- 3$ by $0$ .

$y = 2 + 0$

$x = 0$

Step 4.2.1.2

Add $2$ and $0$ .

$y = 2$

$x = 0$

$y = 2$

$x = 0$

$y = 2$

$x = 0$

$y = 2$

$x = 0$

Step 5

Replace all occurrences of $x$ in $y = 2 - 3 x$ with $\sqrt{3}$ .

$y = 2 - 3 \sqrt{3}$

$x = \sqrt{3}$

Step 6

Replace all occurrences of $x$ with $0$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = 2 - 3 x$ with $0$ .

$y = 2 - 3 \cdot 0$

$x = 0$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $2 - 3 \cdot 0$ .

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Step 6.2.1.1

Multiply $- 3$ by $0$ .

$y = 2 + 0$

$x = 0$

Step 6.2.1.2

Add $2$ and $0$ .

$y = 2$

$x = 0$

$y = 2$

$x = 0$

$y = 2$

$x = 0$

$y = 2$

$x = 0$

Step 7

Replace all occurrences of $x$ in $y = 2 - 3 x$ with $\sqrt{3}$ .

$y = 2 - 3 \sqrt{3}$

$x = \sqrt{3}$

Step 8

Replace all occurrences of $x$ with $- \sqrt{3}$ in each equation.

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Step 8.1

Replace all occurrences of $x$ in $y = 2 - 3 x$ with $- \sqrt{3}$ .

$y = 2 - 3 (- \sqrt{3})$

$x = - \sqrt{3}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Multiply $- 1$ by $- 3$ .

$y = 2 + 3 \sqrt{3}$

$x = - \sqrt{3}$

$y = 2 + 3 \sqrt{3}$

$x = - \sqrt{3}$

$y = 2 + 3 \sqrt{3}$

$x = - \sqrt{3}$

Step 9

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

$(0, 2)$

$(\sqrt{3}, 2 - 3 \sqrt{3})$

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

Step 10

The result can be shown in multiple forms.

Point Form:

$(0, 2), (\sqrt{3}, 2 - 3 \sqrt{3}), (- \sqrt{3}, 2 + 3 \sqrt{3})$

Equation Form:

$x = 0, y = 2$

$x = \sqrt{3}, y = 2 - 3 \sqrt{3}$

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

Step 11