Linear Algebra Examples

$2 x - 6 y = 12$ , $- 3 x + 9 y = - 18$

Step 1

Solve for $x$ in $2 x - 6 y = 12$ .

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Step 1.1

Add $6 y$ to both sides of the equation.

$2 x = 12 + 6 y$

$- 3 x + 9 y = - 18$

Step 1.2

Divide each term in $2 x = 12 + 6 y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 12 + 6 y$ by $2$ .

$\frac{2 x}{2} = \frac{12}{2} + \frac{6 y}{2}$

$- 3 x + 9 y = - 18$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{12}{2} + \frac{6 y}{2}$

$- 3 x + 9 y = - 18$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{2} + \frac{6 y}{2}$

$- 3 x + 9 y = - 18$

$x = \frac{12}{2} + \frac{6 y}{2}$

$- 3 x + 9 y = - 18$

$x = \frac{12}{2} + \frac{6 y}{2}$

$- 3 x + 9 y = - 18$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Divide $12$ by $2$ .

$x = 6 + \frac{6 y}{2}$

$- 3 x + 9 y = - 18$

Step 1.2.3.1.2

Cancel the common factor of $6$ and $2$ .

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Step 1.2.3.1.2.1

Factor $2$ out of $6 y$ .

$x = 6 + \frac{2 (3 y)}{2}$

$- 3 x + 9 y = - 18$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $2$ out of $2$ .

$x = 6 + \frac{2 (3 y)}{2 (1)}$

$- 3 x + 9 y = - 18$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$x = 6 + \frac{2 (3 y)}{2 \cdot 1}$

$- 3 x + 9 y = - 18$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$x = 6 + \frac{3 y}{1}$

$- 3 x + 9 y = - 18$

Step 1.2.3.1.2.2.4

Divide $3 y$ by $1$ .

$x = 6 + 3 y$

$- 3 x + 9 y = - 18$

$x = 6 + 3 y$

$- 3 x + 9 y = - 18$

$x = 6 + 3 y$

$- 3 x + 9 y = - 18$

$x = 6 + 3 y$

$- 3 x + 9 y = - 18$

$x = 6 + 3 y$

$- 3 x + 9 y = - 18$

$x = 6 + 3 y$

$- 3 x + 9 y = - 18$

$x = 6 + 3 y$

$- 3 x + 9 y = - 18$

Step 2

Replace all occurrences of $x$ with $6 + 3 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 3 x + 9 y = - 18$ with $6 + 3 y$ .

$- 3 (6 + 3 y) + 9 y = - 18$

$x = 6 + 3 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 3 (6 + 3 y) + 9 y$ .

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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.

$- 3 \cdot 6 - 3 (3 y) + 9 y = - 18$

$x = 6 + 3 y$

Step 2.2.1.1.2

Multiply $- 3$ by $6$ .

$- 18 - 3 (3 y) + 9 y = - 18$

$x = 6 + 3 y$

Step 2.2.1.1.3

Multiply $3$ by $- 3$ .

$- 18 - 9 y + 9 y = - 18$

$x = 6 + 3 y$

$- 18 - 9 y + 9 y = - 18$

$x = 6 + 3 y$

Step 2.2.1.2

Combine the opposite terms in $- 18 - 9 y + 9 y$ .

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Step 2.2.1.2.1

Add $- 9 y$ and $9 y$ .

$- 18 + 0 = - 18$

$x = 6 + 3 y$

Step 2.2.1.2.2

Add $- 18$ and $0$ .

$- 18 = - 18$

$x = 6 + 3 y$

$- 18 = - 18$

$x = 6 + 3 y$

$- 18 = - 18$

$x = 6 + 3 y$

$- 18 = - 18$

$x = 6 + 3 y$

$- 18 = - 18$

$x = 6 + 3 y$

Step 3

Remove any equations from the system that are always true.

$x = 6 + 3 y$

Step 4