Basic Math Examples

$\frac{y - 2}{5} - \frac{5 - 2 y}{4} = 4 - y - \frac{4 y - 1}{20}$

Step 1

Simplify $\frac{y - 2}{5} - \frac{5 - 2 y}{4}$ .

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Step 1.1

To write $\frac{y - 2}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{y - 2}{5} \cdot \frac{4}{4} - \frac{5 - 2 y}{4} = 4 - y - \frac{4 y - 1}{20}$

Step 1.2

To write $- \frac{5 - 2 y}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{y - 2}{5} \cdot \frac{4}{4} - \frac{5 - 2 y}{4} \cdot \frac{5}{5} = 4 - y - \frac{4 y - 1}{20}$

Step 1.3

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{y - 2}{5}$ by $\frac{4}{4}$ .

$\frac{(y - 2) \cdot 4}{5 \cdot 4} - \frac{5 - 2 y}{4} \cdot \frac{5}{5} = 4 - y - \frac{4 y - 1}{20}$

Step 1.3.2

Multiply $5$ by $4$ .

$\frac{(y - 2) \cdot 4}{20} - \frac{5 - 2 y}{4} \cdot \frac{5}{5} = 4 - y - \frac{4 y - 1}{20}$

Step 1.3.3

Multiply $\frac{5 - 2 y}{4}$ by $\frac{5}{5}$ .

$\frac{(y - 2) \cdot 4}{20} - \frac{(5 - 2 y) \cdot 5}{4 \cdot 5} = 4 - y - \frac{4 y - 1}{20}$

Step 1.3.4

Multiply $4$ by $5$ .

$\frac{(y - 2) \cdot 4}{20} - \frac{(5 - 2 y) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(y - 2) \cdot 4 - (5 - 2 y) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{y \cdot 4 - 2 \cdot 4 - (5 - 2 y) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.2

Move $4$ to the left of $y$ .

$\frac{4 \cdot y - 2 \cdot 4 - (5 - 2 y) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.3

Multiply $- 2$ by $4$ .

$\frac{4 \cdot y - 8 - (5 - 2 y) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.4

Apply the distributive property.

$\frac{4 y - 8 + (- 1 \cdot 5 - (- 2 y)) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.5

Multiply $- 1$ by $5$ .

$\frac{4 y - 8 + (- 5 - (- 2 y)) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.6

Multiply $- 2$ by $- 1$ .

$\frac{4 y - 8 + (- 5 + 2 y) \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.7

Apply the distributive property.

$\frac{4 y - 8 - 5 \cdot 5 + 2 y \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.8

Multiply $- 5$ by $5$ .

$\frac{4 y - 8 - 25 + 2 y \cdot 5}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.9

Multiply $5$ by $2$ .

$\frac{4 y - 8 - 25 + 10 y}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.10

Add $4 y$ and $10 y$ .

$\frac{14 y - 8 - 25}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 1.5.11

Subtract $25$ from $- 8$ .

$\frac{14 y - 33}{20} = 4 - y - \frac{4 y - 1}{20}$

Step 2

Simplify $4 - y - \frac{4 y - 1}{20}$ .

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Step 2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{14 y - 33}{20} = - y + 4 \cdot \frac{20}{20} - \frac{4 y - 1}{20}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $4$ and $\frac{20}{20}$ .

$\frac{14 y - 33}{20} = - y + \frac{4 \cdot 20}{20} - \frac{4 y - 1}{20}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{14 y - 33}{20} = - y + \frac{4 \cdot 20 - (4 y - 1)}{20}$

Step 2.2.3

Multiply $4$ by $20$ .

$\frac{14 y - 33}{20} = - y + \frac{80 - (4 y - 1)}{20}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

$\frac{14 y - 33}{20} = - y + \frac{80 - (4 y) - - 1}{20}$

Step 2.3.2

Multiply $4$ by $- 1$ .

$\frac{14 y - 33}{20} = - y + \frac{80 - 4 y - - 1}{20}$

Step 2.3.3

Multiply $- 1$ by $- 1$ .

$\frac{14 y - 33}{20} = - y + \frac{80 - 4 y + 1}{20}$

Step 2.3.4

Add $80$ and $1$ .

$\frac{14 y - 33}{20} = - y + \frac{- 4 y + 81}{20}$

Step 2.4

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

$\frac{14 y - 33}{20} = - y \cdot \frac{20}{20} + \frac{- 4 y + 81}{20}$

Step 2.5

Simplify terms.

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Step 2.5.1

Combine $- y$ and $\frac{20}{20}$ .

$\frac{14 y - 33}{20} = \frac{- y \cdot 20}{20} + \frac{- 4 y + 81}{20}$

Step 2.5.2

Combine the numerators over the common denominator.

$\frac{14 y - 33}{20} = \frac{- y \cdot 20 - 4 y + 81}{20}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $20$ by $- 1$ .

$\frac{14 y - 33}{20} = \frac{- 20 y - 4 y + 81}{20}$

Step 2.6.2

Subtract $4 y$ from $- 20 y$ .

$\frac{14 y - 33}{20} = \frac{- 24 y + 81}{20}$

Step 2.6.3

Factor $3$ out of $- 24 y + 81$ .

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Step 2.6.3.1

Factor $3$ out of $- 24 y$ .

$\frac{14 y - 33}{20} = \frac{3 (- 8 y) + 81}{20}$

Step 2.6.3.2

Factor $3$ out of $81$ .

$\frac{14 y - 33}{20} = \frac{3 (- 8 y) + 3 (27)}{20}$

Step 2.6.3.3

Factor $3$ out of $3 (- 8 y) + 3 (27)$ .

$\frac{14 y - 33}{20} = \frac{3 (- 8 y + 27)}{20}$

Step 2.7

Simplify with factoring out.

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Step 2.7.1

Factor $- 1$ out of $- 8 y$ .

$\frac{14 y - 33}{20} = \frac{3 (- (8 y) + 27)}{20}$

Step 2.7.2

Rewrite $27$ as $- 1 (- 27)$ .

$\frac{14 y - 33}{20} = \frac{3 (- (8 y) - 1 (- 27))}{20}$

Step 2.7.3

Factor $- 1$ out of $- (8 y) - 1 (- 27)$ .

$\frac{14 y - 33}{20} = \frac{3 (- (8 y - 27))}{20}$

Step 2.7.4

Simplify the expression.

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Step 2.7.4.1

Rewrite $- (8 y - 27)$ as $- 1 (8 y - 27)$ .

$\frac{14 y - 33}{20} = \frac{3 (- 1 (8 y - 27))}{20}$

Step 2.7.4.2

Move the negative in front of the fraction.

$\frac{14 y - 33}{20} = - \frac{3 (8 y - 27)}{20}$

Step 3

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

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Step 3.1

Add $\frac{3 (8 y - 27)}{20}$ to both sides of the equation.

$\frac{14 y - 33}{20} + \frac{3 (8 y - 27)}{20} = 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{14 y - 33 + 3 (8 y - 27)}{20} = 0$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$\frac{14 y - 33 + 3 (8 y) + 3 \cdot - 27}{20} = 0$

Step 3.3.2

Multiply $8$ by $3$ .

$\frac{14 y - 33 + 24 y + 3 \cdot - 27}{20} = 0$

Step 3.3.3

Multiply $3$ by $- 27$ .

$\frac{14 y - 33 + 24 y - 81}{20} = 0$

Step 3.4

Add $14 y$ and $24 y$ .

$\frac{38 y - 33 - 81}{20} = 0$

Step 3.5

Subtract $81$ from $- 33$ .

$\frac{38 y - 114}{20} = 0$

Step 3.6

Cancel the common factor of $38 y - 114$ and $20$ .

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Step 3.6.1

Factor $2$ out of $38 y$ .

$\frac{2 (19 y) - 114}{20} = 0$

Step 3.6.2

Factor $2$ out of $- 114$ .

$\frac{2 (19 y) + 2 (- 57)}{20} = 0$

Step 3.6.3

Factor $2$ out of $2 (19 y) + 2 (- 57)$ .

$\frac{2 (19 y - 57)}{20} = 0$

Step 3.6.4

Cancel the common factors.

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Step 3.6.4.1

Factor $2$ out of $20$ .

$\frac{2 (19 y - 57)}{2 (10)} = 0$

Step 3.6.4.2

Cancel the common factor.

$\frac{2 (19 y - 57)}{2 \cdot 10} = 0$

Step 3.6.4.3

Rewrite the expression.

$\frac{19 y - 57}{10} = 0$

Step 3.7

Factor $19$ out of $19 y - 57$ .

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Step 3.7.1

Factor $19$ out of $19 y$ .

$\frac{19 (y) - 57}{10} = 0$

Step 3.7.2

Factor $19$ out of $- 57$ .

$\frac{19 y + 19 \cdot - 3}{10} = 0$

Step 3.7.3

Factor $19$ out of $19 y + 19 \cdot - 3$ .

$\frac{19 (y - 3)}{10} = 0$

Step 4

Set the numerator equal to zero.

$19 (y - 3) = 0$

Step 5

Solve the equation for $y$ .

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Step 5.1

Divide each term in $19 (y - 3) = 0$ by $19$ and simplify.

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Step 5.1.1

Divide each term in $19 (y - 3) = 0$ by $19$ .

$\frac{19 (y - 3)}{19} = \frac{0}{19}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $19$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{19 (y - 3)}{19} = \frac{0}{19}$

Step 5.1.2.1.2

Divide $y - 3$ by $1$ .

$y - 3 = \frac{0}{19}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $19$ .

$y - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$y = 3$