Linear Algebra Examples

$\frac{5 x}{6} + \frac{3 y}{8} = - 1$

Step 1

Subtract $\frac{5 x}{6}$ from both sides of the equation.

$\frac{3 y}{8} = - 1 - \frac{5 x}{6}$

Step 2

Multiply both sides of the equation by $\frac{8}{3}$ .

$\frac{8}{3} \cdot \frac{3 y}{8} = \frac{8}{3} (- 1 - \frac{5 x}{6})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{8}{3} \cdot \frac{3 y}{8}$ .

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Step 3.1.1.1

Cancel the common factor of $8$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{8}{3} \cdot \frac{3 y}{8} = \frac{8}{3} (- 1 - \frac{5 x}{6})$

Step 3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{8}{3} (- 1 - \frac{5 x}{6})$

Step 3.1.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{8}{3} (- 1 - \frac{5 x}{6})$

Step 3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{8}{3} (- 1 - \frac{5 x}{6})$

Step 3.1.1.2.3

Rewrite the expression.

$y = \frac{8}{3} (- 1 - \frac{5 x}{6})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{8}{3} (- 1 - \frac{5 x}{6})$ .

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Step 3.2.1.1

Apply the distributive property.

$y = \frac{8}{3} \cdot - 1 + \frac{8}{3} (- \frac{5 x}{6})$

Step 3.2.1.2

Multiply $\frac{8}{3} \cdot - 1$ .

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Step 3.2.1.2.1

Combine $\frac{8}{3}$ and $- 1$ .

$y = \frac{8 \cdot - 1}{3} + \frac{8}{3} (- \frac{5 x}{6})$

Step 3.2.1.2.2

Multiply $8$ by $- 1$ .

$y = \frac{- 8}{3} + \frac{8}{3} (- \frac{5 x}{6})$

Step 3.2.1.3

Simplify terms.

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Step 3.2.1.3.1

Cancel the common factor of $2$ .

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Step 3.2.1.3.1.1

Move the leading negative in $- \frac{5 x}{6}$ into the numerator.

$y = \frac{- 8}{3} + \frac{8}{3} \cdot \frac{- 5 x}{6}$

Step 3.2.1.3.1.2

Factor $2$ out of $8$ .

$y = \frac{- 8}{3} + \frac{2 (4)}{3} \cdot \frac{- 5 x}{6}$

Step 3.2.1.3.1.3

Factor $2$ out of $6$ .

$y = \frac{- 8}{3} + \frac{2 \cdot 4}{3} \cdot \frac{- 5 x}{2 \cdot 3}$

Step 3.2.1.3.1.4

Cancel the common factor.

$y = \frac{- 8}{3} + \frac{2 \cdot 4}{3} \cdot \frac{- 5 x}{2 \cdot 3}$

Step 3.2.1.3.1.5

Rewrite the expression.

$y = \frac{- 8}{3} + \frac{4}{3} \cdot \frac{- 5 x}{3}$

Step 3.2.1.3.2

Multiply $\frac{4}{3}$ by $\frac{- 5 x}{3}$ .

$y = \frac{- 8}{3} + \frac{4 (- 5 x)}{3 \cdot 3}$

Step 3.2.1.3.3

Multiply.

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Step 3.2.1.3.3.1

Multiply $- 5$ by $4$ .

$y = \frac{- 8}{3} + \frac{- 20 x}{3 \cdot 3}$

Step 3.2.1.3.3.2

Multiply $3$ by $3$ .

$y = \frac{- 8}{3} + \frac{- 20 x}{9}$

Step 3.2.1.4

Simplify each term.

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Step 3.2.1.4.1

Move the negative in front of the fraction.

$y = - \frac{8}{3} + \frac{- 20 x}{9}$

Step 3.2.1.4.2

Move the negative in front of the fraction.

$y = - \frac{8}{3} - \frac{20 x}{9}$

Step 4

Reorder $- \frac{8}{3}$ and $- \frac{20 x}{9}$ .

$y = - \frac{20 x}{9} - \frac{8}{3}$

Step 5

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 6