Linear Algebra Examples

$\frac{7 a}{3} - (- 1 + 3 a) = 8 + \frac{4 a}{3}$

Step 1

Simplify $\frac{7 a}{3} - (- 1 + 3 a)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{7 a}{3} - - 1 - (3 a) = 8 + \frac{4 a}{3}$

Step 1.1.2

Multiply $- 1$ by $- 1$ .

$\frac{7 a}{3} + 1 - (3 a) = 8 + \frac{4 a}{3}$

Step 1.1.3

Multiply $3$ by $- 1$ .

$\frac{7 a}{3} + 1 - 3 a = 8 + \frac{4 a}{3}$

Step 1.2

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

$\frac{7 a}{3} - 3 a \cdot \frac{3}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $- 3 a$ and $\frac{3}{3}$ .

$\frac{7 a}{3} + \frac{- 3 a \cdot 3}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{7 a - 3 a \cdot 3}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.4

Simplify each term.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $a$ out of $7 a - 3 a \cdot 3$ .

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Step 1.4.1.1.1

Factor $a$ out of $7 a$ .

$\frac{a \cdot 7 - 3 a \cdot 3}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.4.1.1.2

Factor $a$ out of $- 3 a \cdot 3$ .

$\frac{a \cdot 7 + a (- 3 \cdot 3)}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.4.1.1.3

Factor $a$ out of $a \cdot 7 + a (- 3 \cdot 3)$ .

$\frac{a (7 - 3 \cdot 3)}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.4.1.2

Multiply $- 3$ by $3$ .

$\frac{a (7 - 9)}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.4.1.3

Subtract $9$ from $7$ .

$\frac{a \cdot - 2}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.4.2

Move $- 2$ to the left of $a$ .

$\frac{- 2 \cdot a}{3} + 1 = 8 + \frac{4 a}{3}$

Step 1.4.3

Move the negative in front of the fraction.

$- \frac{2 a}{3} + 1 = 8 + \frac{4 a}{3}$

Step 2

Move all terms containing $a$ to the left side of the equation.

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Step 2.1

Subtract $\frac{4 a}{3}$ from both sides of the equation.

$- \frac{2 a}{3} + 1 - \frac{4 a}{3} = 8$

Step 2.2

Combine the numerators over the common denominator.

$1 + \frac{- 2 a - 4 a}{3} = 8$

Step 2.3

Subtract $4 a$ from $- 2 a$ .

$1 + \frac{- 6 a}{3} = 8$

Step 2.4

Cancel the common factor of $- 6$ and $3$ .

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Step 2.4.1

Factor $3$ out of $- 6 a$ .

$1 + \frac{3 (- 2 a)}{3} = 8$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $3$ out of $3$ .

$1 + \frac{3 (- 2 a)}{3 (1)} = 8$

Step 2.4.2.2

Cancel the common factor.

$1 + \frac{3 (- 2 a)}{3 \cdot 1} = 8$

Step 2.4.2.3

Rewrite the expression.

$1 + \frac{- 2 a}{1} = 8$

Step 2.4.2.4

Divide $- 2 a$ by $1$ .

$1 - 2 a = 8$

Step 3

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

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Step 3.1

Subtract $1$ from both sides of the equation.

$- 2 a = 8 - 1$

Step 3.2

Subtract $1$ from $8$ .

$- 2 a = 7$

Step 4

Divide each term in $- 2 a = 7$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 a = 7$ by $- 2$ .

$\frac{- 2 a}{- 2} = \frac{7}{- 2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 a}{- 2} = \frac{7}{- 2}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{7}{- 2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$a = - \frac{7}{2}$

Step 5

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

${a | a = - \frac{7}{2}}$

Step 6