Finite Math Examples

$\frac{6 c}{a - b} = \frac{d}{3 a - f}$

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.

$6 c (3 a - f) = (a - b) d$

Step 2

Solve the equation for $a$ .

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Step 2.1

Simplify $6 c (3 a - f)$ .

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Step 2.1.1

Rewrite.

$0 + 0 + 6 c (3 a - f) = (a - b) d$

Step 2.1.2

Simplify by multiplying through.

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Step 2.1.2.1

Apply the distributive property.

$6 c (3 a) + 6 c (- f) = (a - b) d$

Step 2.1.2.2

Reorder.

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Step 2.1.2.2.1

Rewrite using the commutative property of multiplication.

$6 \cdot 3 c a + 6 c (- f) = (a - b) d$

Step 2.1.2.2.2

Rewrite using the commutative property of multiplication.

$6 \cdot 3 c a + 6 \cdot - 1 c f = (a - b) d$

Step 2.1.3

Simplify each term.

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Step 2.1.3.1

Multiply $6$ by $3$ .

$18 c a + 6 \cdot - 1 c f = (a - b) d$

Step 2.1.3.2

Multiply $6$ by $- 1$ .

$18 c a - 6 c f = (a - b) d$

Step 2.2

Apply the distributive property.

$18 c a - 6 c f = a d - b d$

Step 2.3

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

$18 c a - 6 c f - a d = - b d$

Step 2.4

Add $6 c f$ to both sides of the equation.

$18 c a - a d = - b d + 6 c f$

Step 2.5

Factor $a$ out of $18 c a - a d$ .

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Step 2.5.1

Factor $a$ out of $18 c a$ .

$a (18 c) - a d = - b d + 6 c f$

Step 2.5.2

Factor $a$ out of $- a d$ .

$a (18 c) + a (- 1 d) = - b d + 6 c f$

Step 2.5.3

Factor $a$ out of $a (18 c) + a (- 1 d)$ .

$a (18 c - 1 d) = - b d + 6 c f$

Step 2.6

Rewrite $- 1 d$ as $- d$ .

$a (18 c - d) = - b d + 6 c f$

Step 2.7

Divide each term in $a (18 c - d) = - b d + 6 c f$ by $18 c - d$ and simplify.

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Step 2.7.1

Divide each term in $a (18 c - d) = - b d + 6 c f$ by $18 c - d$ .

$\frac{a (18 c - d)}{18 c - d} = \frac{- b d}{18 c - d} + \frac{6 c f}{18 c - d}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $18 c - d$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{a (18 c - d)}{18 c - d} = \frac{- b d}{18 c - d} + \frac{6 c f}{18 c - d}$

Step 2.7.2.1.2

Divide $a$ by $1$ .

$a = \frac{- b d}{18 c - d} + \frac{6 c f}{18 c - d}$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Combine the numerators over the common denominator.

$a = \frac{- b d + 6 c f}{18 c - d}$

Step 2.7.3.2

Factor $- 1$ out of $- b d$ .

$a = \frac{- (b d) + 6 c f}{18 c - d}$

Step 2.7.3.3

Factor $- 1$ out of $6 c f$ .

$a = \frac{- (b d) - (- 6 c f)}{18 c - d}$

Step 2.7.3.4

Factor $- 1$ out of $- (b d) - (- 6 c f)$ .

$a = \frac{- (b d - 6 c f)}{18 c - d}$

Step 2.7.3.5

Simplify the expression.

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Step 2.7.3.5.1

Rewrite $- (b d - 6 c f)$ as $- 1 (b d - 6 c f)$ .

$a = \frac{- 1 (b d - 6 c f)}{18 c - d}$

Step 2.7.3.5.2

Move the negative in front of the fraction.

$a = - \frac{b d - 6 c f}{18 c - d}$