Linear Algebra Examples

$\frac{3 x + 6}{3} - \frac{2 x - 6}{5} = 2$

Step 1

Simplify $\frac{3 x + 6}{3} - \frac{2 x - 6}{5}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Cancel the common factor of $3 x + 6$ and $3$ .

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Step 1.1.1.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 6}{3} - \frac{2 x - 6}{5} = 2$

Step 1.1.1.2

Factor $3$ out of $6$ .

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

Step 1.1.1.3

Factor $3$ out of $3 (x) + 3 (2)$ .

$\frac{3 (x + 2)}{3} - \frac{2 x - 6}{5} = 2$

Step 1.1.1.4

Cancel the common factors.

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Step 1.1.1.4.1

Factor $3$ out of $3$ .

$\frac{3 (x + 2)}{3 (1)} - \frac{2 x - 6}{5} = 2$

Step 1.1.1.4.2

Cancel the common factor.

$\frac{3 (x + 2)}{3 \cdot 1} - \frac{2 x - 6}{5} = 2$

Step 1.1.1.4.3

Rewrite the expression.

$\frac{x + 2}{1} - \frac{2 x - 6}{5} = 2$

Step 1.1.1.4.4

Divide $x + 2$ by $1$ .

$x + 2 - \frac{2 x - 6}{5} = 2$

Step 1.1.2

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

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Step 1.1.2.1

Factor $2$ out of $2 x$ .

$x + 2 - \frac{2 (x) - 6}{5} = 2$

Step 1.1.2.2

Factor $2$ out of $- 6$ .

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

Step 1.1.2.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$x + 2 - \frac{2 (x - 3)}{5} = 2$

Step 1.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x \cdot \frac{5}{5} - \frac{2 (x - 3)}{5} + 2 = 2$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $x$ and $\frac{5}{5}$ .

$\frac{x \cdot 5}{5} - \frac{2 (x - 3)}{5} + 2 = 2$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{x \cdot 5 - 2 (x - 3)}{5} + 2 = 2$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Move $5$ to the left of $x$ .

$\frac{5 \cdot x - 2 (x - 3)}{5} + 2 = 2$

Step 1.4.2

Apply the distributive property.

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

Step 1.4.3

Multiply $- 2$ by $- 3$ .

$\frac{5 x - 2 x + 6}{5} + 2 = 2$

Step 1.4.4

Subtract $2 x$ from $5 x$ .

$\frac{3 x + 6}{5} + 2 = 2$

Step 1.4.5

Factor $3$ out of $3 x + 6$ .

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Step 1.4.5.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 6}{5} + 2 = 2$

Step 1.4.5.2

Factor $3$ out of $6$ .

$\frac{3 x + 3 \cdot 2}{5} + 2 = 2$

Step 1.4.5.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

$\frac{3 (x + 2)}{5} + 2 = 2$

Step 1.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 (x + 2)}{5} + 2 \cdot \frac{5}{5} = 2$

Step 1.6

Simplify terms.

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Step 1.6.1

Combine $2$ and $\frac{5}{5}$ .

$\frac{3 (x + 2)}{5} + \frac{2 \cdot 5}{5} = 2$

Step 1.6.2

Combine the numerators over the common denominator.

$\frac{3 (x + 2) + 2 \cdot 5}{5} = 2$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 2 + 2 \cdot 5}{5} = 2$

Step 1.7.2

Multiply $3$ by $2$ .

$\frac{3 x + 6 + 2 \cdot 5}{5} = 2$

Step 1.7.3

Multiply $2$ by $5$ .

$\frac{3 x + 6 + 10}{5} = 2$

Step 1.7.4

Add $6$ and $10$ .

$\frac{3 x + 16}{5} = 2$

Step 2

Multiply both sides by $5$ .

$\frac{3 x + 16}{5} \cdot 5 = 2 \cdot 5$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $5$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{3 x + 16}{5} \cdot 5 = 2 \cdot 5$

Step 3.1.1.2

Rewrite the expression.

$3 x + 16 = 2 \cdot 5$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $5$ .

$3 x + 16 = 10$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $16$ from both sides of the equation.

$3 x = 10 - 16$

Step 4.1.2

Subtract $16$ from $10$ .

$3 x = - 6$

Step 4.2

Divide each term in $3 x = - 6$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x = - 6$ by $3$ .

$\frac{3 x}{3} = \frac{- 6}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 6}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 6$ by $3$ .

$x = - 2$

Step 5

The domain is the set of all valid $x$ values.

${x | x = - 2}$

Step 6