Precalculus Examples

$x = 2 y$ , $x = y^{2} - y$

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

Solve $2 y = y^{2} - y$ for $y$ .

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Step 2.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2.2

Move all terms containing $y$ to the left side of the equation.

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Step 2.2.1

Subtract $2 y$ from both sides of the equation.

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

Step 2.2.2

Subtract $2 y$ from $- y$ .

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

Step 2.3

Factor $y$ out of $y^{2} - 3 y$ .

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Step 2.3.1

Factor $y$ out of $y^{2}$ .

$y \cdot y - 3 y = 0$

Step 2.3.2

Factor $y$ out of $- 3 y$ .

$y \cdot y + y \cdot - 3 = 0$

Step 2.3.3

Factor $y$ out of $y \cdot y + y \cdot - 3$ .

$y (y - 3) = 0$

Step 2.4

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

$y = 0$

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

Step 2.5

Set $y$ equal to $0$ .

$y = 0$

Step 2.6

Set $y - 3$ equal to $0$ and solve for $y$ .

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Step 2.6.1

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Step 2.6.2

Add $3$ to both sides of the equation.

$y = 3$

Step 2.7

The final solution is all the values that make $y (y - 3) = 0$ true.

$y = 0, 3$

Step 3

Evaluate $x$ when $y = 0$ .

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Step 3.1

Substitute $0$ for $y$ .

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

Step 3.2

Substitute $0$ for $y$ in $x = {(0)}^{2} - (0)$ and solve for $x$ .

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Step 3.2.1

Remove parentheses.

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

Step 3.2.2

Simplify $0^{2} - (0)$ .

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Step 3.2.2.1

Raising $0$ to any positive power yields $0$ .

$x = 0 - (0)$

Step 3.2.2.2

Subtract $0$ from $0$ .

$x = 0$

Step 4

Evaluate $x$ when $y = 3$ .

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Step 4.1

Substitute $3$ for $y$ .

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

Step 4.2

Substitute $3$ for $y$ in $x = {(3)}^{2} - (3)$ and solve for $x$ .

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Step 4.2.1

Remove parentheses.

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

Step 4.2.2

Simplify $3^{2} - (3)$ .

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

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

$x = 9 - (3)$

Step 4.2.2.1.2

Multiply $- 1$ by $3$ .

$x = 9 - 3$

Step 4.2.2.2

Subtract $3$ from $9$ .

$x = 6$

Step 5

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

$(0, 0)$

$(6, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(0, 0), (6, 3)$

Equation Form:

$x = 0, y = 0$

$x = 6, y = 3$

Step 7