Precalculus Examples

$x^{2} - 4 x + y^{2} + 4 y = 0$ , $x + y = 0$

Step 1

Subtract $y$ from both sides of the equation.

$x = - y$

$x^{2} - 4 x + y^{2} + 4 y = 0$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $x^{2} - 4 x + y^{2} + 4 y = 0$ with $- y$ .

${(- y)}^{2} - 4 (- y) + y^{2} + 4 y = 0$

$x = - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(- y)}^{2} - 4 (- y) + y^{2} + 4 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the product rule to $- y$ .

${(- 1)}^{2} y^{2} - 4 (- y) + y^{2} + 4 y = 0$

$x = - y$

Step 2.2.1.1.2

Raise $- 1$ to the power of $2$ .

$1 y^{2} - 4 (- y) + y^{2} + 4 y = 0$

$x = - y$

Step 2.2.1.1.3

Multiply $y^{2}$ by $1$ .

$y^{2} - 4 (- y) + y^{2} + 4 y = 0$

$x = - y$

Step 2.2.1.1.4

Multiply $- 1$ by $- 4$ .

$y^{2} + 4 y + y^{2} + 4 y = 0$

$x = - y$

$y^{2} + 4 y + y^{2} + 4 y = 0$

$x = - y$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $y^{2}$ and $y^{2}$ .

$2 y^{2} + 4 y + 4 y = 0$

$x = - y$

Step 2.2.1.2.2

Add $4 y$ and $4 y$ .

$2 y^{2} + 8 y = 0$

$x = - y$

$2 y^{2} + 8 y = 0$

$x = - y$

$2 y^{2} + 8 y = 0$

$x = - y$

$2 y^{2} + 8 y = 0$

$x = - y$

$2 y^{2} + 8 y = 0$

$x = - y$

Step 3

Solve for $y$ in $2 y^{2} + 8 y = 0$ .

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Step 3.1

Factor $2 y$ out of $2 y^{2} + 8 y$ .

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Step 3.1.1

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

$2 y (y) + 8 y = 0$

$x = - y$

Step 3.1.2

Factor $2 y$ out of $8 y$ .

$2 y (y) + 2 y (4) = 0$

$x = - y$

Step 3.1.3

Factor $2 y$ out of $2 y (y) + 2 y (4)$ .

$2 y (y + 4) = 0$

$x = - y$

$2 y (y + 4) = 0$

$x = - y$

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 $0$ .

$y = 0$

$y + 4 = 0$

$x = - y$

Step 3.3

Set $y$ equal to $0$ .

$y = 0$

$x = - y$

Step 3.4

Set $y + 4$ equal to $0$ and solve for $y$ .

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Step 3.4.1

Set $y + 4$ equal to $0$ .

$y + 4 = 0$

$x = - y$

Step 3.4.2

Subtract $4$ from both sides of the equation.

$y = - 4$

$x = - y$

$y = - 4$

$x = - y$

Step 3.5

The final solution is all the values that make $2 y (y + 4) = 0$ true.

$y = 0, - 4$

$x = - y$

$y = 0, - 4$

$x = - y$

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 $x = - y$ with $0$ .

$x = - (0)$

$y = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 1$ by $0$ .

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

Step 5

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - y$ with $- 4$ .

$x = - (- 4)$

$y = - 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 1$ by $- 4$ .

$x = 4$

$y = - 4$

$x = 4$

$y = - 4$

$x = 4$

$y = - 4$

Step 6

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

$(0, 0)$

$(4, - 4)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 0), (4, - 4)$

Equation Form:

$x = 0, y = 0$

$x = 4, y = - 4$

Step 8