Linear Algebra Examples

$6 x - 4 y = - 44$ , $2 x + 3 y = - 6$

Step 1

Solve for $x$ in $6 x - 4 y = - 44$ .

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Step 1.1

Add $4 y$ to both sides of the equation.

$6 x = - 44 + 4 y$

$2 x + 3 y = - 6$

Step 1.2

Divide each term in $6 x = - 44 + 4 y$ by $6$ and simplify.

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Step 1.2.1

Divide each term in $6 x = - 44 + 4 y$ by $6$ .

$\frac{6 x}{6} = \frac{- 44}{6} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 44}{6} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 44}{6} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

$x = \frac{- 44}{6} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

$x = \frac{- 44}{6} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

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

Cancel the common factor of $- 44$ and $6$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $- 44$ .

$x = \frac{2 (- 22)}{6} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 22}{2 \cdot 3} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 22}{2 \cdot 3} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 22}{3} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

$x = \frac{- 22}{3} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

$x = \frac{- 22}{3} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{22}{3} + \frac{4 y}{6}$

$2 x + 3 y = - 6$

Step 1.2.3.1.3

Cancel the common factor of $4$ and $6$ .

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Step 1.2.3.1.3.1

Factor $2$ out of $4 y$ .

$x = - \frac{22}{3} + \frac{2 (2 y)}{6}$

$2 x + 3 y = - 6$

Step 1.2.3.1.3.2

Cancel the common factors.

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Step 1.2.3.1.3.2.1

Factor $2$ out of $6$ .

$x = - \frac{22}{3} + \frac{2 (2 y)}{2 (3)}$

$2 x + 3 y = - 6$

Step 1.2.3.1.3.2.2

Cancel the common factor.

$x = - \frac{22}{3} + \frac{2 (2 y)}{2 \cdot 3}$

$2 x + 3 y = - 6$

Step 1.2.3.1.3.2.3

Rewrite the expression.

$x = - \frac{22}{3} + \frac{2 y}{3}$

$2 x + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$2 x + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$2 x + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$2 x + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$2 x + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$2 x + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$2 x + 3 y = - 6$

Step 2

Replace all occurrences of $x$ with $- \frac{22}{3} + \frac{2 y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + 3 y = - 6$ with $- \frac{22}{3} + \frac{2 y}{3}$ .

$2 (- \frac{22}{3} + \frac{2 y}{3}) + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- \frac{22}{3} + \frac{2 y}{3}) + 3 y$ .

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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.

$2 (- \frac{22}{3}) + 2 (\frac{2 y}{3}) + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.1.2

Multiply $2 (- \frac{22}{3})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $2$ .

$- 2 (\frac{22}{3}) + 2 (\frac{2 y}{3}) + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.1.2.2

Combine $- 2$ and $\frac{22}{3}$ .

$\frac{- 2 \cdot 22}{3} + 2 (\frac{2 y}{3}) + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.1.2.3

Multiply $- 2$ by $22$ .

$\frac{- 44}{3} + 2 (\frac{2 y}{3}) + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$\frac{- 44}{3} + 2 (\frac{2 y}{3}) + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.1.3

Multiply $2 \frac{2 y}{3}$ .

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Step 2.2.1.1.3.1

Combine $2$ and $\frac{2 y}{3}$ .

$\frac{- 44}{3} + \frac{2 (2 y)}{3} + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.1.3.2

Multiply $2$ by $2$ .

$\frac{- 44}{3} + \frac{4 y}{3} + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$\frac{- 44}{3} + \frac{4 y}{3} + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.1.4

Move the negative in front of the fraction.

$- \frac{44}{3} + \frac{4 y}{3} + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$- \frac{44}{3} + \frac{4 y}{3} + 3 y = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.2

To write $3 y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{44}{3} + \frac{4 y}{3} + 3 y \cdot \frac{3}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.3

Combine $3 y$ and $\frac{3}{3}$ .

$- \frac{44}{3} + \frac{4 y}{3} + \frac{3 y \cdot 3}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$- \frac{44}{3} + \frac{4 y + 3 y \cdot 3}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{- 44 + 4 y + 3 y \cdot 3}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.6

Multiply $3$ by $3$ .

$\frac{- 44 + 4 y + 9 y}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.7

Add $4 y$ and $9 y$ .

$\frac{- 44 + 13 y}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.8

Rewrite $- 44$ as $- 1 (44)$ .

$\frac{- 1 \cdot 44 + 13 y}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.9

Factor $- 1$ out of $13 y$ .

$\frac{- 1 \cdot 44 - (- 13 y)}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.10

Factor $- 1$ out of $- 1 (44) - (- 13 y)$ .

$\frac{- 1 (44 - 13 y)}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 2.2.1.11

Move the negative in front of the fraction.

$- \frac{44 - 13 y}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$- \frac{44 - 13 y}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$- \frac{44 - 13 y}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$- \frac{44 - 13 y}{3} = - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3

Solve for $y$ in $- \frac{44 - 13 y}{3} = - 6$ .

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Step 3.1

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{44 - 13 y}{3}) = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 3 (- \frac{44 - 13 y}{3})$ .

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Step 3.2.1.1.1

Cancel the common factor of $3$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{44 - 13 y}{3}$ into the numerator.

$- 3 \frac{- (44 - 13 y)}{3} = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.2.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (44 - 13 y)}{3}) = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.2.1.1.1.3

Cancel the common factor.

$3 \cdot (- 1 \frac{- (44 - 13 y)}{3}) = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.2.1.1.1.4

Rewrite the expression.

$44 - 13 y = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$44 - 13 y = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (44 - 13 y) = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.2.1.1.2.2

Multiply $44 - 13 y$ by $1$ .

$44 - 13 y = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$44 - 13 y = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$44 - 13 y = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$44 - 13 y = - 3 \cdot - 6$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $- 3$ by $- 6$ .

$44 - 13 y = 18$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$44 - 13 y = 18$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$44 - 13 y = 18$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.3

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

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Step 3.3.1

Subtract $44$ from both sides of the equation.

$- 13 y = 18 - 44$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.3.2

Subtract $44$ from $18$ .

$- 13 y = - 26$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$- 13 y = - 26$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.4

Divide each term in $- 13 y = - 26$ by $- 13$ and simplify.

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Step 3.4.1

Divide each term in $- 13 y = - 26$ by $- 13$ .

$\frac{- 13 y}{- 13} = \frac{- 26}{- 13}$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 13$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 13 y}{- 13} = \frac{- 26}{- 13}$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 26}{- 13}$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$y = \frac{- 26}{- 13}$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$y = \frac{- 26}{- 13}$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $- 26$ by $- 13$ .

$y = 2$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$y = 2$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$y = 2$

$x = - \frac{22}{3} + \frac{2 y}{3}$

$y = 2$

$x = - \frac{22}{3} + \frac{2 y}{3}$

Step 4

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

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{22}{3} + \frac{2 y}{3}$ with $2$ .

$x = - \frac{22}{3} + \frac{2 (2)}{3}$

$y = 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{22}{3} + \frac{2 (2)}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{- 22 + 2 (2)}{3}$

$y = 2$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $2$ by $2$ .

$x = \frac{- 22 + 4}{3}$

$y = 2$

Step 4.2.1.2.2

Add $- 22$ and $4$ .

$x = \frac{- 18}{3}$

$y = 2$

Step 4.2.1.2.3

Divide $- 18$ by $3$ .

$x = - 6$

$y = 2$

$x = - 6$

$y = 2$

$x = - 6$

$y = 2$

$x = - 6$

$y = 2$

$x = - 6$

$y = 2$

Step 5

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

$(- 6, 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 6, 2)$

Equation Form:

$x = - 6, y = 2$

Step 7