Linear Algebra Examples

$2 x + y = - 2$ , $x + 2 y = 2$

Step 1

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

$y = - 2 - 2 x$

$x + 2 y = 2$

Step 2

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

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Step 2.1

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

$x + 2 (- 2 - 2 x) = 2$

$y = - 2 - 2 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x + 2 (- 2 - 2 x)$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$x + 2 \cdot - 2 + 2 (- 2 x) = 2$

$y = - 2 - 2 x$

Step 2.2.1.1.2

Multiply $2$ by $- 2$ .

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

$y = - 2 - 2 x$

Step 2.2.1.1.3

Multiply $- 2$ by $2$ .

$x - 4 - 4 x = 2$

$y = - 2 - 2 x$

$x - 4 - 4 x = 2$

$y = - 2 - 2 x$

Step 2.2.1.2

Subtract $4 x$ from $x$ .

$- 3 x - 4 = 2$

$y = - 2 - 2 x$

$- 3 x - 4 = 2$

$y = - 2 - 2 x$

$- 3 x - 4 = 2$

$y = - 2 - 2 x$

$- 3 x - 4 = 2$

$y = - 2 - 2 x$

Step 3

Solve for $x$ in $- 3 x - 4 = 2$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Add $4$ to both sides of the equation.

$- 3 x = 2 + 4$

$y = - 2 - 2 x$

Step 3.1.2

Add $2$ and $4$ .

$- 3 x = 6$

$y = - 2 - 2 x$

$- 3 x = 6$

$y = - 2 - 2 x$

Step 3.2

Divide each term in $- 3 x = 6$ by $- 3$ and simplify.

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Step 3.2.1

Divide each term in $- 3 x = 6$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{6}{- 3}$

$y = - 2 - 2 x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{6}{- 3}$

$y = - 2 - 2 x$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{- 3}$

$y = - 2 - 2 x$

$x = \frac{6}{- 3}$

$y = - 2 - 2 x$

$x = \frac{6}{- 3}$

$y = - 2 - 2 x$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $6$ by $- 3$ .

$x = - 2$

$y = - 2 - 2 x$

$x = - 2$

$y = - 2 - 2 x$

$x = - 2$

$y = - 2 - 2 x$

$x = - 2$

$y = - 2 - 2 x$

Step 4

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

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Step 4.1

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

$y = - 2 - 2 \cdot - 2$

$x = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 2 - 2 \cdot - 2$ .

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Step 4.2.1.1

Multiply $- 2$ by $- 2$ .

$y = - 2 + 4$

$x = - 2$

Step 4.2.1.2

Add $- 2$ and $4$ .

$y = 2$

$x = - 2$

$y = 2$

$x = - 2$

$y = 2$

$x = - 2$

$y = 2$

$x = - 2$

Step 5

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

$(- 2, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, 2)$

Equation Form:

$x = - 2, y = 2$

Step 7

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