Finite Math Examples

$y - \frac{2 y - 7}{\frac{7}{\frac{20}{21} + \frac{4}{3 y}}}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$y - ((2 y - 7) \frac{\frac{20}{21} + \frac{4}{3 y}}{7})$

Step 1.2

Simplify the numerator.

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Step 1.2.1

To write $\frac{20}{21}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y - ((2 y - 7) \frac{\frac{20}{21} \cdot \frac{y}{y} + \frac{4}{3 y}}{7})$

Step 1.2.2

To write $\frac{4}{3 y}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y - ((2 y - 7) \frac{\frac{20}{21} \cdot \frac{y}{y} + \frac{4}{3 y} \cdot \frac{7}{7}}{7})$

Step 1.2.3

Write each expression with a common denominator of $21 y$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.3.1

Multiply $\frac{20}{21}$ by $\frac{y}{y}$ .

$y - ((2 y - 7) \frac{\frac{20 y}{21 y} + \frac{4}{3 y} \cdot \frac{7}{7}}{7})$

Step 1.2.3.2

Multiply $\frac{4}{3 y}$ by $\frac{7}{7}$ .

$y - ((2 y - 7) \frac{\frac{20 y}{21 y} + \frac{4 \cdot 7}{3 y \cdot 7}}{7})$

Step 1.2.3.3

Multiply $7$ by $3$ .

$y - ((2 y - 7) \frac{\frac{20 y}{21 y} + \frac{4 \cdot 7}{21 y}}{7})$

Step 1.2.4

Combine the numerators over the common denominator.

$y - ((2 y - 7) \frac{\frac{20 y + 4 \cdot 7}{21 y}}{7})$

Step 1.2.5

Factor $4$ out of $20 y + 4 \cdot 7$ .

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Step 1.2.5.1

Factor $4$ out of $20 y$ .

$y - ((2 y - 7) \frac{\frac{4 (5 y) + 4 \cdot 7}{21 y}}{7})$

Step 1.2.5.2

Factor $4$ out of $4 \cdot 7$ .

$y - ((2 y - 7) \frac{\frac{4 (5 y) + 4 (7)}{21 y}}{7})$

Step 1.2.5.3

Factor $4$ out of $4 (5 y) + 4 (7)$ .

$y - ((2 y - 7) \frac{\frac{4 (5 y + 7)}{21 y}}{7})$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$y - ((2 y - 7) (\frac{4 (5 y + 7)}{21 y} \cdot \frac{1}{7}))$

Step 1.4

Multiply $\frac{4 (5 y + 7)}{21 y} \cdot \frac{1}{7}$ .

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Step 1.4.1

Multiply $\frac{4 (5 y + 7)}{21 y}$ by $\frac{1}{7}$ .

$y - ((2 y - 7) \frac{4 (5 y + 7)}{21 y \cdot 7})$

Step 1.4.2

Multiply $7$ by $21$ .

$y - ((2 y - 7) \frac{4 (5 y + 7)}{147 y})$

Step 1.5

Multiply $2 y - 7$ by $\frac{4 (5 y + 7)}{147 y}$ .

$y - \frac{(2 y - 7) (4 (5 y + 7))}{147 y}$

Step 1.6

Move $4$ to the left of $2 y - 7$ .

$y - \frac{4 (2 y - 7) (5 y + 7)}{147 y}$

Step 2

To write $y$ as a fraction with a common denominator, multiply by $\frac{147 y}{147 y}$ .

$y \cdot \frac{147 y}{147 y} - \frac{4 (2 y - 7) (5 y + 7)}{147 y}$

Step 3

Simplify terms.

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Step 3.1

Combine $y$ and $\frac{147 y}{147 y}$ .

$\frac{y (147 y)}{147 y} - \frac{4 (2 y - 7) (5 y + 7)}{147 y}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{y (147 y) - 4 (2 y - 7) (5 y + 7)}{147 y}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite using the commutative property of multiplication.

$\frac{147 y \cdot y - 4 (2 y - 7) (5 y + 7)}{147 y}$

Step 4.2

Multiply $y$ by $y$ by adding the exponents.

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Step 4.2.1

Move $y$ .

$\frac{147 (y \cdot y) - 4 (2 y - 7) (5 y + 7)}{147 y}$

Step 4.2.2

Multiply $y$ by $y$ .

$\frac{147 y^{2} - 4 (2 y - 7) (5 y + 7)}{147 y}$

Step 4.3

Apply the distributive property.

$\frac{147 y^{2} + (- 4 (2 y) - 4 \cdot - 7) (5 y + 7)}{147 y}$

Step 4.4

Multiply $2$ by $- 4$ .

$\frac{147 y^{2} + (- 8 y - 4 \cdot - 7) (5 y + 7)}{147 y}$

Step 4.5

Multiply $- 4$ by $- 7$ .

$\frac{147 y^{2} + (- 8 y + 28) (5 y + 7)}{147 y}$

Step 4.6

Expand $(- 8 y + 28) (5 y + 7)$ using the FOIL Method.

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Step 4.6.1

Apply the distributive property.

$\frac{147 y^{2} - 8 y (5 y + 7) + 28 (5 y + 7)}{147 y}$

Step 4.6.2

Apply the distributive property.

$\frac{147 y^{2} - 8 y (5 y) - 8 y \cdot 7 + 28 (5 y + 7)}{147 y}$

Step 4.6.3

Apply the distributive property.

$\frac{147 y^{2} - 8 y (5 y) - 8 y \cdot 7 + 28 (5 y) + 28 \cdot 7}{147 y}$

Step 4.7

Simplify and combine like terms.

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Step 4.7.1

Simplify each term.

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Step 4.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{147 y^{2} - 8 \cdot 5 y \cdot y - 8 y \cdot 7 + 28 (5 y) + 28 \cdot 7}{147 y}$

Step 4.7.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 4.7.1.2.1

Move $y$ .

$\frac{147 y^{2} - 8 \cdot 5 (y \cdot y) - 8 y \cdot 7 + 28 (5 y) + 28 \cdot 7}{147 y}$

Step 4.7.1.2.2

Multiply $y$ by $y$ .

$\frac{147 y^{2} - 8 \cdot 5 y^{2} - 8 y \cdot 7 + 28 (5 y) + 28 \cdot 7}{147 y}$

Step 4.7.1.3

Multiply $- 8$ by $5$ .

$\frac{147 y^{2} - 40 y^{2} - 8 y \cdot 7 + 28 (5 y) + 28 \cdot 7}{147 y}$

Step 4.7.1.4

Multiply $7$ by $- 8$ .

$\frac{147 y^{2} - 40 y^{2} - 56 y + 28 (5 y) + 28 \cdot 7}{147 y}$

Step 4.7.1.5

Multiply $5$ by $28$ .

$\frac{147 y^{2} - 40 y^{2} - 56 y + 140 y + 28 \cdot 7}{147 y}$

Step 4.7.1.6

Multiply $28$ by $7$ .

$\frac{147 y^{2} - 40 y^{2} - 56 y + 140 y + 196}{147 y}$

Step 4.7.2

Add $- 56 y$ and $140 y$ .

$\frac{147 y^{2} - 40 y^{2} + 84 y + 196}{147 y}$

Step 4.8

Subtract $40 y^{2}$ from $147 y^{2}$ .

$\frac{107 y^{2} + 84 y + 196}{147 y}$