Finite Math Examples

$\frac{x - 3}{4} = \frac{15}{14} - \frac{x + 5}{7}$

Step 1

Multiply both sides by $4$ .

$\frac{x - 3}{4} \cdot 4 = (\frac{15}{14} - \frac{x + 5}{7}) \cdot 4$

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 $4$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{x - 3}{4} \cdot 4 = (\frac{15}{14} - \frac{x + 5}{7}) \cdot 4$

Step 2.1.1.2

Rewrite the expression.

$x - 3 = (\frac{15}{14} - \frac{x + 5}{7}) \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{15}{14} - \frac{x + 5}{7}) \cdot 4$ .

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Step 2.2.1.1

To write $- \frac{x + 5}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x - 3 = (\frac{15}{14} - \frac{x + 5}{7} \cdot \frac{2}{2}) \cdot 4$

Step 2.2.1.2

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.2.1

Multiply $\frac{x + 5}{7}$ by $\frac{2}{2}$ .

$x - 3 = (\frac{15}{14} - \frac{(x + 5) \cdot 2}{7 \cdot 2}) \cdot 4$

Step 2.2.1.2.2

Multiply $7$ by $2$ .

$x - 3 = (\frac{15}{14} - \frac{(x + 5) \cdot 2}{14}) \cdot 4$

Step 2.2.1.3

Combine the numerators over the common denominator.

$x - 3 = \frac{15 - (x + 5) \cdot 2}{14} \cdot 4$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Apply the distributive property.

$x - 3 = \frac{15 + (- x - 1 \cdot 5) \cdot 2}{14} \cdot 4$

Step 2.2.1.4.2

Multiply $- 1$ by $5$ .

$x - 3 = \frac{15 + (- x - 5) \cdot 2}{14} \cdot 4$

Step 2.2.1.4.3

Apply the distributive property.

$x - 3 = \frac{15 - x \cdot 2 - 5 \cdot 2}{14} \cdot 4$

Step 2.2.1.4.4

Multiply $2$ by $- 1$ .

$x - 3 = \frac{15 - 2 x - 5 \cdot 2}{14} \cdot 4$

Step 2.2.1.4.5

Multiply $- 5$ by $2$ .

$x - 3 = \frac{15 - 2 x - 10}{14} \cdot 4$

Step 2.2.1.4.6

Subtract $10$ from $15$ .

$x - 3 = \frac{- 2 x + 5}{14} \cdot 4$

Step 2.2.1.5

Simplify terms.

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Step 2.2.1.5.1

Cancel the common factor of $2$ .

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Step 2.2.1.5.1.1

Factor $2$ out of $14$ .

$x - 3 = \frac{- 2 x + 5}{2 (7)} \cdot 4$

Step 2.2.1.5.1.2

Factor $2$ out of $4$ .

$x - 3 = \frac{- 2 x + 5}{2 \cdot 7} \cdot (2 \cdot 2)$

Step 2.2.1.5.1.3

Cancel the common factor.

$x - 3 = \frac{- 2 x + 5}{2 \cdot 7} \cdot (2 \cdot 2)$

Step 2.2.1.5.1.4

Rewrite the expression.

$x - 3 = \frac{- 2 x + 5}{7} \cdot 2$

Step 2.2.1.5.2

Combine $\frac{- 2 x + 5}{7}$ and $2$ .

$x - 3 = \frac{(- 2 x + 5) \cdot 2}{7}$

Step 2.2.1.5.3

Move $2$ to the left of $- 2 x + 5$ .

$x - 3 = \frac{2 \cdot (- 2 x + 5)}{7}$

Step 2.2.1.5.4

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

$x - 3 = \frac{2 (- (2 x) + 5)}{7}$

Step 2.2.1.5.5

Rewrite $5$ as $- 1 (- 5)$ .

$x - 3 = \frac{2 (- (2 x) - 1 (- 5))}{7}$

Step 2.2.1.5.6

Factor $- 1$ out of $- (2 x) - 1 (- 5)$ .

$x - 3 = \frac{2 (- (2 x - 5))}{7}$

Step 2.2.1.5.7

Simplify the expression.

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Step 2.2.1.5.7.1

Rewrite $- (2 x - 5)$ as $- 1 (2 x - 5)$ .

$x - 3 = \frac{2 (- 1 (2 x - 5))}{7}$

Step 2.2.1.5.7.2

Move the negative in front of the fraction.

$x - 3 = - \frac{2 (2 x - 5)}{7}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $7$ .

$(x - 3) \cdot 7 = - \frac{2 (2 x - 5)}{7} \cdot 7$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(x - 3) \cdot 7$ .

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Step 3.2.1.1.1

Apply the distributive property.

$x \cdot 7 - 3 \cdot 7 = - \frac{2 (2 x - 5)}{7} \cdot 7$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $7$ to the left of $x$ .

$7 \cdot x - 3 \cdot 7 = - \frac{2 (2 x - 5)}{7} \cdot 7$

Step 3.2.1.1.2.2

Multiply $- 3$ by $7$ .

$7 x - 21 = - \frac{2 (2 x - 5)}{7} \cdot 7$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{2 (2 x - 5)}{7} \cdot 7$ .

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Step 3.2.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{2 (2 x - 5)}{7}$ into the numerator.

$7 x - 21 = \frac{- 2 (2 x - 5)}{7} \cdot 7$

Step 3.2.2.1.1.2

Cancel the common factor.

$7 x - 21 = \frac{- 2 (2 x - 5)}{7} \cdot 7$

Step 3.2.2.1.1.3

Rewrite the expression.

$7 x - 21 = - 2 (2 x - 5)$

Step 3.2.2.1.2

Apply the distributive property.

$7 x - 21 = - 2 (2 x) - 2 \cdot - 5$

Step 3.2.2.1.3

Multiply.

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Step 3.2.2.1.3.1

Multiply $2$ by $- 2$ .

$7 x - 21 = - 4 x - 2 \cdot - 5$

Step 3.2.2.1.3.2

Multiply $- 2$ by $- 5$ .

$7 x - 21 = - 4 x + 10$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Add $4 x$ to both sides of the equation.

$7 x - 21 + 4 x = 10$

Step 3.3.1.2

Add $7 x$ and $4 x$ .

$11 x - 21 = 10$

Step 3.3.2

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

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Step 3.3.2.1

Add $21$ to both sides of the equation.

$11 x = 10 + 21$

Step 3.3.2.2

Add $10$ and $21$ .

$11 x = 31$

Step 3.3.3

Divide each term in $11 x = 31$ by $11$ and simplify.

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Step 3.3.3.1

Divide each term in $11 x = 31$ by $11$ .

$\frac{11 x}{11} = \frac{31}{11}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $11$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{31}{11}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{31}{11}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{31}{11}$

Decimal Form:

$x = 2.\overline{81}$

Mixed Number Form:

$x = 2 \frac{9}{11}$