Finite Math Examples

$\frac{2 x + b}{3 a - b} = \frac{x - a}{b - 3 a}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(2 x + b) (b - 3 a) = (3 a - b) (x - a)$

Step 2

Solve the equation for $x$ .

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Step 2.1

Simplify $(2 x + b) (b - 3 a)$ .

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Step 2.1.1

Rewrite.

$0 + 0 + (2 x + b) (b - 3 a) = (3 a - b) (x - a)$

Step 2.1.2

Simplify by adding zeros.

$(2 x + b) (b - 3 a) = (3 a - b) (x - a)$

Step 2.1.3

Expand $(2 x + b) (b - 3 a)$ using the FOIL Method.

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Step 2.1.3.1

Apply the distributive property.

$2 x (b - 3 a) + b (b - 3 a) = (3 a - b) (x - a)$

Step 2.1.3.2

Apply the distributive property.

$2 x b + 2 x (- 3 a) + b (b - 3 a) = (3 a - b) (x - a)$

Step 2.1.3.3

Apply the distributive property.

$2 x b + 2 x (- 3 a) + b \cdot b + b (- 3 a) = (3 a - b) (x - a)$

Step 2.1.4

Simplify each term.

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Step 2.1.4.1

Rewrite using the commutative property of multiplication.

$2 x b + 2 \cdot - 3 x a + b \cdot b + b (- 3 a) = (3 a - b) (x - a)$

Step 2.1.4.2

Multiply $2$ by $- 3$ .

$2 x b - 6 x a + b \cdot b + b (- 3 a) = (3 a - b) (x - a)$

Step 2.1.4.3

Multiply $b$ by $b$ .

$2 x b - 6 x a + b^{2} + b (- 3 a) = (3 a - b) (x - a)$

Step 2.1.4.4

Rewrite using the commutative property of multiplication.

$2 x b - 6 x a + b^{2} - 3 b a = (3 a - b) (x - a)$

Step 2.2

Simplify $(3 a - b) (x - a)$ .

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Step 2.2.1

Expand $(3 a - b) (x - a)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$2 x b - 6 x a + b^{2} - 3 b a = 3 a (x - a) - b (x - a)$

Step 2.2.1.2

Apply the distributive property.

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x + 3 a (- a) - b (x - a)$

Step 2.2.1.3

Apply the distributive property.

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x + 3 a (- a) - b x - b (- a)$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Rewrite using the commutative property of multiplication.

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x + 3 \cdot - 1 a \cdot a - b x - b (- a)$

Step 2.2.2.2

Multiply $a$ by $a$ by adding the exponents.

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Step 2.2.2.2.1

Move $a$ .

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x + 3 \cdot - 1 (a \cdot a) - b x - b (- a)$

Step 2.2.2.2.2

Multiply $a$ by $a$ .

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x + 3 \cdot - 1 a^{2} - b x - b (- a)$

Step 2.2.2.3

Multiply $3$ by $- 1$ .

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x - 3 a^{2} - b x - b (- a)$

Step 2.2.2.4

Rewrite using the commutative property of multiplication.

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x - 3 a^{2} - b x - 1 \cdot - 1 b a$

Step 2.2.2.5

Multiply $- 1$ by $- 1$ .

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x - 3 a^{2} - b x + 1 b a$

Step 2.2.2.6

Multiply $b$ by $1$ .

$2 x b - 6 x a + b^{2} - 3 b a = 3 a x - 3 a^{2} - b x + b a$

Step 2.3

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

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Step 2.3.1

Subtract $3 a x$ from both sides of the equation.

$2 x b - 6 x a + b^{2} - 3 b a - 3 a x = - 3 a^{2} - b x + b a$

Step 2.3.2

Add $b x$ to both sides of the equation.

$2 x b - 6 x a + b^{2} - 3 b a - 3 a x + b x = - 3 a^{2} + b a$

Step 2.3.3

Add $2 x b$ and $b x$ .

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Step 2.3.3.1

Move $x$ .

$- 6 x a + b^{2} - 3 b a - 3 a x + 2 b x + b x = - 3 a^{2} + b a$

Step 2.3.3.2

Add $2 b x$ and $b x$ .

$- 6 x a + b^{2} - 3 b a - 3 a x + 3 b x = - 3 a^{2} + b a$

Step 2.3.4

Subtract $3 a x$ from $- 6 x a$ .

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Step 2.3.4.1

Move $x$ .

$b^{2} - 3 b a - 6 a x - 3 a x + 3 b x = - 3 a^{2} + b a$

Step 2.3.4.2

Subtract $3 a x$ from $- 6 a x$ .

$b^{2} - 3 b a - 9 a x + 3 b x = - 3 a^{2} + b a$

Step 2.4

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

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Step 2.4.1

Subtract $b^{2}$ from both sides of the equation.

$- 3 b a - 9 a x + 3 b x = - 3 a^{2} + b a - b^{2}$

Step 2.4.2

Add $3 b a$ to both sides of the equation.

$- 9 a x + 3 b x = - 3 a^{2} + b a - b^{2} + 3 b a$

Step 2.4.3

Add $b a$ and $3 b a$ .

$- 9 a x + 3 b x = - 3 a^{2} - b^{2} + 4 b a$

Step 2.5

Factor $3 x$ out of $- 9 a x + 3 b x$ .

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Step 2.5.1

Factor $3 x$ out of $- 9 a x$ .

$3 x (- 3 a) + 3 b x = - 3 a^{2} - b^{2} + 4 b a$

Step 2.5.2

Factor $3 x$ out of $3 b x$ .

$3 x (- 3 a) + 3 x (b) = - 3 a^{2} - b^{2} + 4 b a$

Step 2.5.3

Factor $3 x$ out of $3 x (- 3 a) + 3 x (b)$ .

$3 x (- 3 a + b) = - 3 a^{2} - b^{2} + 4 b a$

Step 2.6

Divide each term in $3 x (- 3 a + b) = - 3 a^{2} - b^{2} + 4 b a$ by $3 (- 3 a + b)$ and simplify.

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Step 2.6.1

Divide each term in $3 x (- 3 a + b) = - 3 a^{2} - b^{2} + 4 b a$ by $3 (- 3 a + b)$ .

$\frac{3 x (- 3 a + b)}{3 (- 3 a + b)} = \frac{- 3 a^{2}}{3 (- 3 a + b)} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $3$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{3 x (- 3 a + b)}{3 (- 3 a + b)} = \frac{- 3 a^{2}}{3 (- 3 a + b)} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.2.1.2

Rewrite the expression.

$\frac{x (- 3 a + b)}{- 3 a + b} = \frac{- 3 a^{2}}{3 (- 3 a + b)} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.2.2

Cancel the common factor of $- 3 a + b$ .

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Step 2.6.2.2.1

Cancel the common factor.

$\frac{x (- 3 a + b)}{- 3 a + b} = \frac{- 3 a^{2}}{3 (- 3 a + b)} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3 a^{2}}{3 (- 3 a + b)} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Simplify each term.

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Step 2.6.3.1.1

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

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Step 2.6.3.1.1.1

Factor $3$ out of $- 3 a^{2}$ .

$x = \frac{3 (- a^{2})}{3 (- 3 a + b)} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.1.1.2

Cancel the common factors.

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Step 2.6.3.1.1.2.1

Cancel the common factor.

$x = \frac{3 (- a^{2})}{3 (- 3 a + b)} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.1.1.2.2

Rewrite the expression.

$x = \frac{- a^{2}}{- 3 a + b} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.1.2

Move the negative in front of the fraction.

$x = - \frac{a^{2}}{- 3 a + b} + \frac{- b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.1.3

Move the negative in front of the fraction.

$x = - \frac{a^{2}}{- 3 a + b} - \frac{b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.2

To write $- \frac{a^{2}}{- 3 a + b}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{a^{2}}{- 3 a + b} \cdot \frac{3}{3} - \frac{b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.3

Write each expression with a common denominator of $(- 3 a + b) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.3.3.1

Multiply $\frac{a^{2}}{- 3 a + b}$ by $\frac{3}{3}$ .

$x = - \frac{a^{2} \cdot 3}{(- 3 a + b) \cdot 3} - \frac{b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.3.2

Reorder the factors of $(- 3 a + b) \cdot 3$ .

$x = - \frac{a^{2} \cdot 3}{3 (- 3 a + b)} - \frac{b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.4

Combine the numerators over the common denominator.

$x = \frac{- a^{2} \cdot 3 - b^{2}}{3 (- 3 a + b)} + \frac{4 b a}{3 (- 3 a + b)}$

Step 2.6.3.5

Combine the numerators over the common denominator.

$x = \frac{- a^{2} \cdot 3 - b^{2} + 4 b a}{3 (- 3 a + b)}$

Step 2.6.3.6

Multiply $3$ by $- 1$ .

$x = \frac{- 3 a^{2} - b^{2} + 4 b a}{3 (- 3 a + b)}$

Step 2.6.3.7

Factor by grouping.

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Step 2.6.3.7.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot - 1 = 3$ and whose sum is $b = 4$ .

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Step 2.6.3.7.1.1

Reorder terms.

$x = \frac{- 3 a^{2} - b^{2} + 4 a b}{3 (- 3 a + b)}$

Step 2.6.3.7.1.2

Reorder $- b^{2}$ and $4 a b$ .

$x = \frac{- 3 a^{2} + 4 a b - b^{2}}{3 (- 3 a + b)}$

Step 2.6.3.7.1.3

Factor $4$ out of $4 a b$ .

$x = \frac{- 3 a^{2} + 4 (a b) - b^{2}}{3 (- 3 a + b)}$

Step 2.6.3.7.1.4

Rewrite $4$ as $1$ plus $3$

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

Step 2.6.3.7.1.5

Apply the distributive property.

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

Step 2.6.3.7.1.6

Multiply $a b$ by $1$ .

$x = \frac{- 3 a^{2} + a b + 3 (a b) - b^{2}}{3 (- 3 a + b)}$

Step 2.6.3.7.1.7

Move parentheses.

$x = \frac{- 3 a^{2} + a b + 3 a b - b^{2}}{3 (- 3 a + b)}$

Step 2.6.3.7.2

Factor out the greatest common factor from each group.

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Step 2.6.3.7.2.1

Group the first two terms and the last two terms.

$x = \frac{(- 3 a^{2} + a b) + 3 a b - b^{2}}{3 (- 3 a + b)}$

Step 2.6.3.7.2.2

Factor out the greatest common factor (GCF) from each group.

$x = \frac{a (- 3 a + b) - b (- 3 a + b)}{3 (- 3 a + b)}$

Step 2.6.3.7.3

Factor the polynomial by factoring out the greatest common factor, $- 3 a + b$ .

$x = \frac{(- 3 a + b) (a - b)}{3 (- 3 a + b)}$

Step 2.6.3.8

Cancel the common factor of $- 3 a + b$ .

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Step 2.6.3.8.1

Cancel the common factor.

$x = \frac{(- 3 a + b) (a - b)}{3 (- 3 a + b)}$

Step 2.6.3.8.2

Rewrite the expression.

$x = \frac{a - b}{3}$