Solve for y (y-1)/4-(y-4)/5=(y+4)/10-1

$\frac{y - 1}{4} - \frac{y - 4}{5} = \frac{y + 4}{10} - 1$

Step 1

Simplify $\frac{y - 1}{4} - \frac{y - 4}{5}$ .

To write $\frac{y - 1}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{y - 1}{4} \cdot \frac{5}{5} - \frac{y - 4}{5} = \frac{y + 4}{10} - 1$

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

$\frac{y - 1}{4} \cdot \frac{5}{5} - \frac{y - 4}{5} \cdot \frac{4}{4} = \frac{y + 4}{10} - 1$

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

Multiply $\frac{y - 1}{4}$ by $\frac{5}{5}$ .

$\frac{(y - 1) \cdot 5}{4 \cdot 5} - \frac{y - 4}{5} \cdot \frac{4}{4} = \frac{y + 4}{10} - 1$

Multiply $4$ by $5$ .

$\frac{(y - 1) \cdot 5}{20} - \frac{y - 4}{5} \cdot \frac{4}{4} = \frac{y + 4}{10} - 1$

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

$\frac{(y - 1) \cdot 5}{20} - \frac{(y - 4) \cdot 4}{5 \cdot 4} = \frac{y + 4}{10} - 1$

Multiply $5$ by $4$ .

$\frac{(y - 1) \cdot 5}{20} - \frac{(y - 4) \cdot 4}{20} = \frac{y + 4}{10} - 1$

Combine the numerators over the common denominator.

$\frac{(y - 1) \cdot 5 - (y - 4) \cdot 4}{20} = \frac{y + 4}{10} - 1$

Simplify the numerator.

Apply the distributive property.

$\frac{y \cdot 5 - 1 \cdot 5 - (y - 4) \cdot 4}{20} = \frac{y + 4}{10} - 1$

Move $5$ to the left of $y$ .

$\frac{5 \cdot y - 1 \cdot 5 - (y - 4) \cdot 4}{20} = \frac{y + 4}{10} - 1$

Multiply $- 1$ by $5$ .

$\frac{5 \cdot y - 5 - (y - 4) \cdot 4}{20} = \frac{y + 4}{10} - 1$

Apply the distributive property.

$\frac{5 y - 5 + (- y - - 4) \cdot 4}{20} = \frac{y + 4}{10} - 1$

Multiply $- 1$ by $- 4$ .

$\frac{5 y - 5 + (- y + 4) \cdot 4}{20} = \frac{y + 4}{10} - 1$

Apply the distributive property.

$\frac{5 y - 5 - y \cdot 4 + 4 \cdot 4}{20} = \frac{y + 4}{10} - 1$

Multiply $4$ by $- 1$ .

$\frac{5 y - 5 - 4 y + 4 \cdot 4}{20} = \frac{y + 4}{10} - 1$

Multiply $4$ by $4$ .

$\frac{5 y - 5 - 4 y + 16}{20} = \frac{y + 4}{10} - 1$

Subtract $4 y$ from $5 y$ .

$\frac{y - 5 + 16}{20} = \frac{y + 4}{10} - 1$

Add $- 5$ and $16$ .

$\frac{y + 11}{20} = \frac{y + 4}{10} - 1$

Step 2

Simplify $\frac{y + 4}{10} - 1$ .

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{y + 11}{20} = \frac{y + 4}{10} - 1 \cdot \frac{10}{10}$

Simplify terms.

Combine $- 1$ and $\frac{10}{10}$ .

$\frac{y + 11}{20} = \frac{y + 4}{10} + \frac{- 1 \cdot 10}{10}$

Combine the numerators over the common denominator.

$\frac{y + 11}{20} = \frac{y + 4 - 1 \cdot 10}{10}$

Simplify the numerator.

Multiply $- 1$ by $10$ .

$\frac{y + 11}{20} = \frac{y + 4 - 10}{10}$

Subtract $10$ from $4$ .

$\frac{y + 11}{20} = \frac{y - 6}{10}$

Step 3

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

Subtract $\frac{y - 6}{10}$ from both sides of the equation.

$\frac{y + 11}{20} - \frac{y - 6}{10} = 0$

To write $- \frac{y - 6}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y + 11}{20} - \frac{y - 6}{10} \cdot \frac{2}{2} = 0$

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

Multiply $\frac{y - 6}{10}$ by $\frac{2}{2}$ .

$\frac{y + 11}{20} - \frac{(y - 6) \cdot 2}{10 \cdot 2} = 0$

Multiply $10$ by $2$ .

$\frac{y + 11}{20} - \frac{(y - 6) \cdot 2}{20} = 0$

Combine the numerators over the common denominator.

$\frac{y + 11 - (y - 6) \cdot 2}{20} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{y + 11 + (- y - - 6) \cdot 2}{20} = 0$

Multiply $- 1$ by $- 6$ .

$\frac{y + 11 + (- y + 6) \cdot 2}{20} = 0$

Apply the distributive property.

$\frac{y + 11 - y \cdot 2 + 6 \cdot 2}{20} = 0$

Multiply $2$ by $- 1$ .

$\frac{y + 11 - 2 y + 6 \cdot 2}{20} = 0$

Multiply $6$ by $2$ .

$\frac{y + 11 - 2 y + 12}{20} = 0$

Subtract $2 y$ from $y$ .

$\frac{- y + 11 + 12}{20} = 0$

Add $11$ and $12$ .

$\frac{- y + 23}{20} = 0$

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

$\frac{- (y) + 23}{20} = 0$

Rewrite $23$ as $- 1 (- 23)$ .

$\frac{- (y) - 1 (- 23)}{20} = 0$

Factor $- 1$ out of $- (y) - 1 (- 23)$ .

$\frac{- (y - 23)}{20} = 0$

Rewrite $- (y - 23)$ as $- 1 (y - 23)$ .

$\frac{- 1 (y - 23)}{20} = 0$

Move the negative in front of the fraction.

$- \frac{y - 23}{20} = 0$

Step 4

Set the numerator equal to zero.

$y - 23 = 0$

Step 5

Add $23$ to both sides of the equation.

$y = 23$

Solve for y (y-1)/4-(y-4)/5=(y+4)/10-1

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