Trigonometry Examples

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

Step 1

Add $x$ to both sides of the equation.

$y = 2 + x$

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

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 $x^{2} + y^{2} = 4$ with $2 + x$ .

$x^{2} + {(2 + x)}^{2} = 4$

$y = 2 + x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} + {(2 + x)}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(2 + x)}^{2}$ as $(2 + x) (2 + x)$ .

$x^{2} + (2 + x) (2 + x) = 4$

$y = 2 + x$

Step 2.2.1.1.2

Expand $(2 + x) (2 + x)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$x^{2} + 2 (2 + x) + x (2 + x) = 4$

$y = 2 + x$

Step 2.2.1.1.2.2

Apply the distributive property.

$x^{2} + 2 \cdot 2 + 2 x + x (2 + x) = 4$

$y = 2 + x$

Step 2.2.1.1.2.3

Apply the distributive property.

$x^{2} + 2 \cdot 2 + 2 x + x \cdot 2 + x \cdot x = 4$

$y = 2 + x$

$x^{2} + 2 \cdot 2 + 2 x + x \cdot 2 + x \cdot x = 4$

$y = 2 + x$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $2$ by $2$ .

$x^{2} + 4 + 2 x + x \cdot 2 + x \cdot x = 4$

$y = 2 + x$

Step 2.2.1.1.3.1.2

Move $2$ to the left of $x$ .

$x^{2} + 4 + 2 x + 2 \cdot x + x \cdot x = 4$

$y = 2 + x$

Step 2.2.1.1.3.1.3

Multiply $x$ by $x$ .

$x^{2} + 4 + 2 x + 2 x + x^{2} = 4$

$y = 2 + x$

$x^{2} + 4 + 2 x + 2 x + x^{2} = 4$

$y = 2 + x$

Step 2.2.1.1.3.2

Add $2 x$ and $2 x$ .

$x^{2} + 4 + 4 x + x^{2} = 4$

$y = 2 + x$

$x^{2} + 4 + 4 x + x^{2} = 4$

$y = 2 + x$

$x^{2} + 4 + 4 x + x^{2} = 4$

$y = 2 + x$

Step 2.2.1.2

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

$2 x^{2} + 4 + 4 x = 4$

$y = 2 + x$

$2 x^{2} + 4 + 4 x = 4$

$y = 2 + x$

$2 x^{2} + 4 + 4 x = 4$

$y = 2 + x$

$2 x^{2} + 4 + 4 x = 4$

$y = 2 + x$

Step 3

Solve for $x$ in $2 x^{2} + 4 + 4 x = 4$ .

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Step 3.1

Subtract $4$ from both sides of the equation.

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

$y = 2 + x$

Step 3.2

Combine the opposite terms in $2 x^{2} + 4 + 4 x - 4$ .

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Step 3.2.1

Subtract $4$ from $4$ .

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

$y = 2 + x$

Step 3.2.2

Add $2 x^{2} + 4 x$ and $0$ .

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

$y = 2 + x$

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

$y = 2 + x$

Step 3.3

Factor $2 x$ out of $2 x^{2} + 4 x$ .

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Step 3.3.1

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

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

$y = 2 + x$

Step 3.3.2

Factor $2 x$ out of $4 x$ .

$2 x (x) + 2 x (2) = 0$

$y = 2 + x$

Step 3.3.3

Factor $2 x$ out of $2 x (x) + 2 x (2)$ .

$2 x (x + 2) = 0$

$y = 2 + x$

$2 x (x + 2) = 0$

$y = 2 + x$

Step 3.4

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 3.5

Set $x$ equal to $0$ .

$x = 0$

$y = 2 + x$

Step 3.6

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

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Step 3.6.1

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

$x + 2 = 0$

$y = 2 + x$

Step 3.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

$y = 2 + x$

$x = - 2$

$y = 2 + x$

Step 3.7

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

$y = 2 + 0$

$x = 0$

Step 4.2

Simplify $y = 2 + 0$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$y = 2 + 0$

$x = 0$

$y = 2 + 0$

$x = 0$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

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 $x$ with $- 2$ in each equation.

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Step 5.1

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

$y = 2 - 2$

$x = - 2$

Step 5.2

Simplify $y = 2 - 2$ .

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Remove parentheses.

$y = 2 - 2$

$x = - 2$

$y = 2 - 2$

$x = - 2$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Subtract $2$ from $2$ .

$y = 0$

$x = - 2$

$y = 0$

$x = - 2$

$y = 0$

$x = - 2$

$y = 0$

$x = - 2$

Step 6

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

$(0, 2)$

$(- 2, 0)$

Step 7

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

$x = 0, y = 2$

$x = - 2, y = 0$

Step 8