Precalculus Examples

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

Step 1

Add $2 y$ to both sides of the equation.

$x = 2 y$

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

Step 2

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

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Step 2.1

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

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

$x = 2 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply $2$ by $3$ .

$6 y - y^{2} = 0$

$x = 2 y$

$6 y - y^{2} = 0$

$x = 2 y$

$6 y - y^{2} = 0$

$x = 2 y$

Step 3

Solve for $y$ in $6 y - y^{2} = 0$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$6 u - u^{2} = 0$

$x = 2 y$

Step 3.1.2

Factor $u$ out of $6 u - u^{2}$ .

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Step 3.1.2.1

Factor $u$ out of $6 u$ .

$u \cdot 6 - u^{2} = 0$

$x = 2 y$

Step 3.1.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 6 + u (- u) = 0$

$x = 2 y$

Step 3.1.2.3

Factor $u$ out of $u \cdot 6 + u (- u)$ .

$u \cdot (6 - u) = 0$

$x = 2 y$

Step 3.1.2.4

Multiply $u$ by $6 - u$ .

$u (6 - u) = 0$

$x = 2 y$

$u (6 - u) = 0$

$x = 2 y$

Step 3.1.3

Replace all occurrences of $u$ with $y$ .

$y (6 - y) = 0$

$x = 2 y$

$y (6 - y) = 0$

$x = 2 y$

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 $0$ .

$y = 0$

$6 - y = 0$

$x = 2 y$

Step 3.3

Set $y$ equal to $0$ .

$y = 0$

$x = 2 y$

Step 3.4

Set $6 - y$ equal to $0$ and solve for $y$ .

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Step 3.4.1

Set $6 - y$ equal to $0$ .

$6 - y = 0$

$x = 2 y$

Step 3.4.2

Solve $6 - y = 0$ for $y$ .

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Step 3.4.2.1

Subtract $6$ from both sides of the equation.

$- y = - 6$

$x = 2 y$

Step 3.4.2.2

Divide each term in $- y = - 6$ by $- 1$ and simplify.

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Step 3.4.2.2.1

Divide each term in $- y = - 6$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 6}{- 1}$

$x = 2 y$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 6}{- 1}$

$x = 2 y$

Step 3.4.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 6}{- 1}$

$x = 2 y$

$y = \frac{- 6}{- 1}$

$x = 2 y$

Step 3.4.2.2.3

Simplify the right side.

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Step 3.4.2.2.3.1

Divide $- 6$ by $- 1$ .

$y = 6$

$x = 2 y$

$y = 6$

$x = 2 y$

$y = 6$

$x = 2 y$

$y = 6$

$x = 2 y$

$y = 6$

$x = 2 y$

Step 3.5

The final solution is all the values that make $y (6 - y) = 0$ true.

$y = 0, 6$

$x = 2 y$

$y = 0, 6$

$x = 2 y$

Step 4

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

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Step 4.1

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

$x = 2 (0)$

$y = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $2$ by $0$ .

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

Step 5

Replace all occurrences of $y$ with $6$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = 2 y$ with $6$ .

$x = 2 (6)$

$y = 6$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $2$ by $6$ .

$x = 12$

$y = 6$

$x = 12$

$y = 6$

$x = 12$

$y = 6$

Step 6

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

$(0, 0)$

$(12, 6)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 0), (12, 6)$

Equation Form:

$x = 0, y = 0$

$x = 12, y = 6$

Step 8