Finite Math Examples

$\frac{2 x}{3 - c} = a x + 1$

Step 1

Multiply both sides by $3 - c$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $3 - c$ .

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

$2 x = (a x + 1) (3 - c)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(a x + 1) (3 - c)$ .

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Step 2.2.1.1

Expand $(a x + 1) (3 - c)$ using the FOIL Method.

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Step 2.2.1.1.1

Apply the distributive property.

$2 x = a x (3 - c) + 1 (3 - c)$

Step 2.2.1.1.2

Apply the distributive property.

$2 x = a x \cdot 3 + a x (- c) + 1 (3 - c)$

Step 2.2.1.1.3

Apply the distributive property.

$2 x = a x \cdot 3 + a x (- c) + 1 \cdot 3 + 1 (- c)$

Step 2.2.1.2

Simplify terms.

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Step 2.2.1.2.1

Simplify each term.

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Step 2.2.1.2.1.1

Move $3$ to the left of $a x$ .

$2 x = 3 \cdot (a x) + a x (- c) + 1 \cdot 3 + 1 (- c)$

Step 2.2.1.2.1.2

Rewrite using the commutative property of multiplication.

$2 x = 3 a x - a x c + 1 \cdot 3 + 1 (- c)$

Step 2.2.1.2.1.3

Multiply $3$ by $1$ .

$2 x = 3 a x - a x c + 3 + 1 (- c)$

Step 2.2.1.2.1.4

Multiply $- c$ by $1$ .

$2 x = 3 a x - a x c + 3 - c$

Step 2.2.1.2.2

Simplify the expression.

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Step 2.2.1.2.2.1

Move $x$ .

$2 x = 3 a x - a c x + 3 - c$

Step 2.2.1.2.2.2

Move $3$ .

$2 x = 3 a x - a c x - c + 3$

Step 2.2.1.2.2.3

Reorder $3 a x$ and $- a c x$ .

$2 x = - a c x + 3 a x - c + 3$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- a c x + 3 a x - c + 3 = 2 x$

Step 3.2

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

$- a c x + 3 a x - c + 3 - 2 x = 0$

Step 3.3

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

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Step 3.3.1

Add $c$ to both sides of the equation.

$- a c x + 3 a x + 3 - 2 x = c$

Step 3.3.2

Subtract $3$ from both sides of the equation.

$- a c x + 3 a x - 2 x = c - 3$

Step 3.4

Factor $x$ out of $- a c x + 3 a x - 2 x$ .

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Step 3.4.1

Factor $x$ out of $- a c x$ .

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Factor $x$ out of $x (- a c + 3 a) + x \cdot - 2$ .

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

Step 3.5

Divide each term in $x (- a c + 3 a - 2) = c - 3$ by $- a c + 3 a - 2$ and simplify.

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Step 3.5.1

Divide each term in $x (- a c + 3 a - 2) = c - 3$ by $- a c + 3 a - 2$ .

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

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $- a c + 3 a - 2$ .

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Step 3.5.2.1.1

Cancel the common factor.

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

Step 3.5.2.1.2

Divide $x$ by $1$ .

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

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Combine the numerators over the common denominator.

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

Step 3.5.3.2

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

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

Step 3.5.3.3

Factor $- 1$ out of $3 a$ .

$x = \frac{c - 3}{- (a c) - (- 3 a) - 2}$

Step 3.5.3.4

Factor $- 1$ out of $- (a c) - (- 3 a)$ .

$x = \frac{c - 3}{- (a c - 3 a) - 2}$

Step 3.5.3.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{c - 3}{- (a c - 3 a) - 1 (2)}$

Step 3.5.3.6

Factor $- 1$ out of $- (a c - 3 a) - 1 (2)$ .

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

Step 3.5.3.7

Rewrite negatives.

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Step 3.5.3.7.1

Rewrite $- (a c - 3 a + 2)$ as $- 1 (a c - 3 a + 2)$ .

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

Step 3.5.3.7.2

Move the negative in front of the fraction.

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