Linear Algebra Examples

$x - 3 y = 13$ , $2 x - 2 y = 6$

Step 1

Add $3 y$ to both sides of the equation.

$x = 13 + 3 y$

$2 x - 2 y = 6$

Step 2

Replace all occurrences of $x$ with $13 + 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x - 2 y = 6$ with $13 + 3 y$ .

$2 (13 + 3 y) - 2 y = 6$

$x = 13 + 3 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (13 + 3 y) - 2 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$2 \cdot 13 + 2 (3 y) - 2 y = 6$

$x = 13 + 3 y$

Step 2.2.1.1.2

Multiply $2$ by $13$ .

$26 + 2 (3 y) - 2 y = 6$

$x = 13 + 3 y$

Step 2.2.1.1.3

Multiply $3$ by $2$ .

$26 + 6 y - 2 y = 6$

$x = 13 + 3 y$

$26 + 6 y - 2 y = 6$

$x = 13 + 3 y$

Step 2.2.1.2

Subtract $2 y$ from $6 y$ .

$26 + 4 y = 6$

$x = 13 + 3 y$

$26 + 4 y = 6$

$x = 13 + 3 y$

$26 + 4 y = 6$

$x = 13 + 3 y$

$26 + 4 y = 6$

$x = 13 + 3 y$

Step 3

Solve for $y$ in $26 + 4 y = 6$ .

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Step 3.1

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

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Step 3.1.1

Subtract $26$ from both sides of the equation.

$4 y = 6 - 26$

$x = 13 + 3 y$

Step 3.1.2

Subtract $26$ from $6$ .

$4 y = - 20$

$x = 13 + 3 y$

$4 y = - 20$

$x = 13 + 3 y$

Step 3.2

Divide each term in $4 y = - 20$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 y = - 20$ by $4$ .

$\frac{4 y}{4} = \frac{- 20}{4}$

$x = 13 + 3 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{- 20}{4}$

$x = 13 + 3 y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 20}{4}$

$x = 13 + 3 y$

$y = \frac{- 20}{4}$

$x = 13 + 3 y$

$y = \frac{- 20}{4}$

$x = 13 + 3 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 20$ by $4$ .

$y = - 5$

$x = 13 + 3 y$

$y = - 5$

$x = 13 + 3 y$

$y = - 5$

$x = 13 + 3 y$

$y = - 5$

$x = 13 + 3 y$

Step 4

Replace all occurrences of $y$ with $- 5$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 13 + 3 y$ with $- 5$ .

$x = 13 + 3 (- 5)$

$y = - 5$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $13 + 3 (- 5)$ .

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Step 4.2.1.1

Multiply $3$ by $- 5$ .

$x = 13 - 15$

$y = - 5$

Step 4.2.1.2

Subtract $15$ from $13$ .

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

$x = - 2$

$y = - 5$

Step 5

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

$(- 2, - 5)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 2, - 5)$

Equation Form:

$x = - 2, y = - 5$

Step 7