Precalculus Examples

x + y = 0

$x + y = 0$ ,

x - y = 0

$x - y = 0$

Step 1

Solve the system of equations.

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Step 1.1

Multiply each equation by the value that makes the coefficients of

x

$x$ opposite.

x + y = 0

$x + y = 0$

(- 1) \cdot (x - y) = (- 1) (0)

$(- 1) \cdot (x - y) = (- 1) (0)$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Simplify

(- 1) \cdot (x - y)

$(- 1) \cdot (x - y)$ .

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Step 1.2.1.1.1

Apply the distributive property.

x + y = 0

$x + y = 0$

- 1 x - 1 (- y) = (- 1) (0)

$- 1 x - 1 (- y) = (- 1) (0)$

Step 1.2.1.1.2

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

x + y = 0

$x + y = 0$

- x - 1 (- y) = (- 1) (0)

$- x - 1 (- y) = (- 1) (0)$

Step 1.2.1.1.3

Multiply

- 1 (- y)

$- 1 (- y)$ .

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Step 1.2.1.1.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

x + y = 0

$x + y = 0$

- x + 1 y = (- 1) (0)

$- x + 1 y = (- 1) (0)$

Step 1.2.1.1.3.2

Multiply

y

$y$ by

1

$1$ .

x + y = 0

$x + y = 0$

- x + y = (- 1) (0)

$- x + y = (- 1) (0)$

x + y = 0

$x + y = 0$

- x + y = (- 1) (0)

$- x + y = (- 1) (0)$

x + y = 0

$x + y = 0$

- x + y = (- 1) (0)

$- x + y = (- 1) (0)$

x + y = 0

$x + y = 0$

- x + y = (- 1) (0)

$- x + y = (- 1) (0)$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Multiply

- 1

$- 1$ by

0

$0$ .

x + y = 0

$x + y = 0$

- x + y = 0

$- x + y = 0$

x + y = 0

$x + y = 0$

- x + y = 0

$- x + y = 0$

x + y = 0

$x + y = 0$

- x + y = 0

$- x + y = 0$

Step 1.3

Add the two equations together to eliminate

x

$x$ from the system.

		$x$ $x$	$+$ $+$	$y$ $y$	$=$ $=$	$0$ $0$
$+$ $+$	$-$ $-$	$x$ $x$	$+$ $+$	$y$ $y$	$=$ $=$	$0$ $0$
			$2$ $2$	$y$ $y$	$=$ $=$	$0$ $0$

Step 1.4

Divide each term in

2 y = 0

$2 y = 0$ by

2

$2$ and simplify.

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Step 1.4.1

Divide each term in

2 y = 0

$2 y = 0$ by

2

$2$ .

\frac{2 y}{2} = \frac{0}{2}

$\frac{2 y}{2} = \frac{0}{2}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of

2

$2$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{0}{2}$

Step 1.4.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{0}{2}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide

$0$ by

$2$ .

$y = 0$

Step 1.5

Substitute the value found for

$y$ into one of the original equations, then solve for

$x$ .

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Step 1.5.1

Substitute the value found for

$y$ into one of the original equations to solve for

$x$ .

$x + 0 = 0$

Step 1.5.2

Add

$x$ and

$0$ .

$x = 0$

Step 1.6

The solution to the independent system of equations can be represented as a point.

$(0, 0)$

Step 2

Since the system has a point of intersection, the system is independent.

Independent

Step 3

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