Finite Math Examples

$(2 - \sqrt{3}) a + (1 - \sqrt{3}) b = 7 - \sqrt{3}$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$2 a - \sqrt{3} a + (1 - \sqrt{3}) b = 7 - \sqrt{3}$

Step 1.2

Apply the distributive property.

$2 a - \sqrt{3} a + 1 b - \sqrt{3} b = 7 - \sqrt{3}$

Step 1.3

Multiply $b$ by $1$ .

$2 a - \sqrt{3} a + b - \sqrt{3} b = 7 - \sqrt{3}$

Step 2

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

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Step 2.1

Subtract $2 a$ from both sides of the equation.

$- \sqrt{3} a + b - \sqrt{3} b = 7 - \sqrt{3} - 2 a$

Step 2.2

Add $\sqrt{3} a$ to both sides of the equation.

$b - \sqrt{3} b = 7 - \sqrt{3} - 2 a + \sqrt{3} a$

Step 3

Factor $b$ out of $b - \sqrt{3} b$ .

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Step 3.1

Factor $b$ out of $b^{1}$ .

$b \cdot 1 - \sqrt{3} b = 7 - \sqrt{3} - 2 a + \sqrt{3} a$

Step 3.2

Factor $b$ out of $- \sqrt{3} b$ .

$b \cdot 1 + b (- \sqrt{3}) = 7 - \sqrt{3} - 2 a + \sqrt{3} a$

Step 3.3

Factor $b$ out of $b \cdot 1 + b (- \sqrt{3})$ .

$b (1 - \sqrt{3}) = 7 - \sqrt{3} - 2 a + \sqrt{3} a$

Step 4

Divide each term in $b (1 - \sqrt{3}) = 7 - \sqrt{3} - 2 a + \sqrt{3} a$ by $1 - \sqrt{3}$ and simplify.

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Step 4.1

Divide each term in $b (1 - \sqrt{3}) = 7 - \sqrt{3} - 2 a + \sqrt{3} a$ by $1 - \sqrt{3}$ .

$\frac{b (1 - \sqrt{3})}{1 - \sqrt{3}} = \frac{7}{1 - \sqrt{3}} + \frac{- \sqrt{3}}{1 - \sqrt{3}} + \frac{- 2 a}{1 - \sqrt{3}} + \frac{\sqrt{3} a}{1 - \sqrt{3}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $1 - \sqrt{3}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{b (1 - \sqrt{3})}{1 - \sqrt{3}} = \frac{7}{1 - \sqrt{3}} + \frac{- \sqrt{3}}{1 - \sqrt{3}} + \frac{- 2 a}{1 - \sqrt{3}} + \frac{\sqrt{3} a}{1 - \sqrt{3}}$

Step 4.2.1.2

Divide $b$ by $1$ .

$b = \frac{7}{1 - \sqrt{3}} + \frac{- \sqrt{3}}{1 - \sqrt{3}} + \frac{- 2 a}{1 - \sqrt{3}} + \frac{\sqrt{3} a}{1 - \sqrt{3}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$b = \frac{7 - \sqrt{3} - 2 a + \sqrt{3} a}{1 - \sqrt{3}}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Reorder terms.

$b = \frac{- 2 a + a \sqrt{3} + 7 - \sqrt{3}}{1 - \sqrt{3}}$

Step 4.3.2.2

Rewrite $- 2 a + a \sqrt{3} + 7 - \sqrt{3}$ in a factored form.

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Step 4.3.2.2.1

Regroup terms.

$b = \frac{- \sqrt{3} - 2 a + a \sqrt{3} + 7}{1 - \sqrt{3}}$

Step 4.3.2.2.2

Factor $- 1$ out of $- 2 a + a \sqrt{3} + 7$ .

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Step 4.3.2.2.2.1

Factor $- 1$ out of $- 2 a$ .

$b = \frac{- \sqrt{3} - (2 a) + a \sqrt{3} + 7}{1 - \sqrt{3}}$

Step 4.3.2.2.2.2

Factor $- 1$ out of $a \sqrt{3}$ .

$b = \frac{- \sqrt{3} - (2 a) - (- a \sqrt{3}) + 7}{1 - \sqrt{3}}$

Step 4.3.2.2.2.3

Rewrite $7$ as $- 1 (- 7)$ .

$b = \frac{- \sqrt{3} - (2 a) - (- a \sqrt{3}) - 1 (- 7)}{1 - \sqrt{3}}$

Step 4.3.2.2.2.4

Factor $- 1$ out of $- (2 a) - (- a \sqrt{3})$ .

$b = \frac{- \sqrt{3} - (2 a - a \sqrt{3}) - 1 (- 7)}{1 - \sqrt{3}}$

Step 4.3.2.2.2.5

Factor $- 1$ out of $- (2 a - a \sqrt{3}) - 1 (- 7)$ .

$b = \frac{- \sqrt{3} - (2 a - a \sqrt{3} - 7)}{1 - \sqrt{3}}$

Step 4.3.2.2.3

Factor $- 1$ out of $- \sqrt{3} - (2 a - a \sqrt{3} - 7)$ .

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Step 4.3.2.2.3.1

Reorder $- \sqrt{3}$ and $- (2 a - a \sqrt{3} - 7)$ .

$b = \frac{- (2 a - a \sqrt{3} - 7) - \sqrt{3}}{1 - \sqrt{3}}$

Step 4.3.2.2.3.2

Factor $- 1$ out of $- \sqrt{3}$ .

$b = \frac{- (2 a - a \sqrt{3} - 7) - (\sqrt{3})}{1 - \sqrt{3}}$

Step 4.3.2.2.3.3

Factor $- 1$ out of $- (2 a - a \sqrt{3} - 7) - (\sqrt{3})$ .

$b = \frac{- (2 a - a \sqrt{3} - 7 + \sqrt{3})}{1 - \sqrt{3}}$

Step 4.3.3

Move the negative in front of the fraction.

$b = - \frac{2 a - a \sqrt{3} - 7 + \sqrt{3}}{1 - \sqrt{3}}$

Step 4.3.4

Multiply $\frac{2 a - a \sqrt{3} - 7 + \sqrt{3}}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

$b = - (\frac{2 a - a \sqrt{3} - 7 + \sqrt{3}}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}})$

Step 4.3.5

Multiply $\frac{2 a - a \sqrt{3} - 7 + \sqrt{3}}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

$b = - \frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{(1 - \sqrt{3}) (1 + \sqrt{3})}$

Step 4.3.6

Expand the denominator using the FOIL method.

$b = - \frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{1 + \sqrt{3} - \sqrt{3} - {\sqrt{3}}^{2}}$

Step 4.3.7

Simplify.

$b = - \frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{- 2}$

Step 4.3.8

Move the negative in front of the fraction.

$b = - - \frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{2}$

Step 4.3.9

Multiply $- - \frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{2}$ .

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Step 4.3.9.1

Multiply $- 1$ by $- 1$ .

$b = 1 \frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{2}$

Step 4.3.9.2

Multiply $\frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{2}$ by $1$ .

$b = \frac{(2 a - a \sqrt{3} - 7 + \sqrt{3}) (1 + \sqrt{3})}{2}$