Precalculus Examples

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

Step 1

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

$y = - 2 x$

$x^{2} - y = 0$

Step 2

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

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Step 2.1

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

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

$y = - 2 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply $- 2$ by $- 1$ .

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

$y = - 2 x$

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

$y = - 2 x$

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

$y = - 2 x$

Step 3

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

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Step 3.1

Factor $x$ out of $x^{2} + 2 x$ .

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Step 3.1.1

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

$x \cdot x + 2 x = 0$

$y = - 2 x$

Step 3.1.2

Factor $x$ out of $2 x$ .

$x \cdot x + x \cdot 2 = 0$

$y = - 2 x$

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$x (x + 2) = 0$

$y = - 2 x$

$x (x + 2) = 0$

$y = - 2 x$

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 $0$ .

$x = 0$

$x + 2 = 0$

$y = - 2 x$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

$y = - 2 x$

Step 3.4

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

$y = - 2 x$

Step 3.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

$y = - 2 x$

$x = - 2$

$y = - 2 x$

Step 3.5

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

$x = 0, - 2$

$y = - 2 x$

$x = 0, - 2$

$y = - 2 x$

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 $y = - 2 x$ with $0$ .

$y = - 2 \cdot 0$

$x = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 2$ by $0$ .

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

Step 5

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

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Step 5.1

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

$y = - 2 \cdot - 2$

$x = - 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 2$ by $- 2$ .

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

$y = 4$

$x = - 2$

Step 6

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

$(0, 0)$

$(- 2, 4)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(0, 0), (- 2, 4)$

Equation Form:

$x = 0, y = 0$

$x = - 2, y = 4$

Step 8