Linear Algebra Examples

$4 x + 11 = - 6 y$ , $6 x + 9 y = - 3$

Step 1

Solve for $x$ in $4 x + 11 = - 6 y$ .

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Step 1.1

Subtract $11$ from both sides of the equation.

$4 x = - 6 y - 11$

$6 x + 9 y = - 3$

Step 1.2

Divide each term in $4 x = - 6 y - 11$ by $4$ and simplify.

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Step 1.2.1

Divide each term in $4 x = - 6 y - 11$ by $4$ .

$\frac{4 x}{4} = \frac{- 6 y}{4} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 6 y}{4} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6 y}{4} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

$x = \frac{- 6 y}{4} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

$x = \frac{- 6 y}{4} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

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 $- 6$ and $4$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $- 6 y$ .

$x = \frac{2 (- 3 y)}{4} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

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 $4$ .

$x = \frac{2 (- 3 y)}{2 (2)} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 (- 3 y)}{2 \cdot 2} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 3 y}{2} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

$x = \frac{- 3 y}{2} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

$x = \frac{- 3 y}{2} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{3 y}{2} + \frac{- 11}{4}$

$6 x + 9 y = - 3$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$x = - \frac{3 y}{2} - \frac{11}{4}$

$6 x + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$6 x + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$6 x + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$6 x + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$6 x + 9 y = - 3$

Step 2

Replace all occurrences of $x$ with $- \frac{3 y}{2} - \frac{11}{4}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $6 x + 9 y = - 3$ with $- \frac{3 y}{2} - \frac{11}{4}$ .

$6 (- \frac{3 y}{2} - \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $6 (- \frac{3 y}{2} - \frac{11}{4}) + 9 y$ .

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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.

$6 (- \frac{3 y}{2}) + 6 (- \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{3 y}{2}$ into the numerator.

$6 (\frac{- 3 y}{2}) + 6 (- \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.2.2

Factor $2$ out of $6$ .

$2 (3) (\frac{- 3 y}{2}) + 6 (- \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.2.3

Cancel the common factor.

$2 \cdot (3 (\frac{- 3 y}{2})) + 6 (- \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.2.4

Rewrite the expression.

$3 (- 3 y) + 6 (- \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$3 (- 3 y) + 6 (- \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.3

Multiply $- 3$ by $3$ .

$- 9 y + 6 (- \frac{11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{11}{4}$ into the numerator.

$- 9 y + 6 (\frac{- 11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.4.2

Factor $2$ out of $6$ .

$- 9 y + 2 (3) (\frac{- 11}{4}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.4.3

Factor $2$ out of $4$ .

$- 9 y + 2 \cdot (3 (\frac{- 11}{2 \cdot 2})) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.4.4

Cancel the common factor.

$- 9 y + 2 \cdot (3 (\frac{- 11}{2 \cdot 2})) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.4.5

Rewrite the expression.

$- 9 y + 3 (\frac{- 11}{2}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$- 9 y + 3 (\frac{- 11}{2}) + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.5

Combine $3$ and $\frac{- 11}{2}$ .

$- 9 y + \frac{3 \cdot - 11}{2} + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.6

Multiply $3$ by $- 11$ .

$- 9 y + \frac{- 33}{2} + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.1.7

Move the negative in front of the fraction.

$- 9 y - \frac{33}{2} + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$- 9 y - \frac{33}{2} + 9 y = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.2

Combine the opposite terms in $- 9 y - \frac{33}{2} + 9 y$ .

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Step 2.2.1.2.1

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

$0 - \frac{33}{2} = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 2.2.1.2.2

Subtract $\frac{33}{2}$ from $0$ .

$- \frac{33}{2} = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$- \frac{33}{2} = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$- \frac{33}{2} = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$- \frac{33}{2} = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

$- \frac{33}{2} = - 3$

$x = - \frac{3 y}{2} - \frac{11}{4}$

Step 3

Since $- \frac{33}{2} = - 3$ is not true, there is no solution.

No solution

Step 4