Solve for y 1/3(y-2)-5/6(y+1)=3/4(y-3)-2

$\frac{1}{3} (y - 2) - \frac{5}{6} (y + 1) = \frac{3}{4} (y - 3) - 2$

Step 1

Simplify $\frac{1}{3} \cdot (y - 2) - \frac{5}{6} \cdot (y + 1)$ .

Simplify each term.

Apply the distributive property.

$\frac{1}{3} y + \frac{1}{3} \cdot - 2 - \frac{5}{6} \cdot (y + 1) = \frac{3}{4} \cdot (y - 3) - 2$

Combine $\frac{1}{3}$ and $y$ .

$\frac{y}{3} + \frac{1}{3} \cdot - 2 - \frac{5}{6} \cdot (y + 1) = \frac{3}{4} \cdot (y - 3) - 2$

Combine $\frac{1}{3}$ and $- 2$ .

$\frac{y}{3} + \frac{- 2}{3} - \frac{5}{6} \cdot (y + 1) = \frac{3}{4} \cdot (y - 3) - 2$

Move the negative in front of the fraction.

$\frac{y}{3} - \frac{2}{3} - \frac{5}{6} \cdot (y + 1) = \frac{3}{4} \cdot (y - 3) - 2$

Apply the distributive property.

$\frac{y}{3} - \frac{2}{3} - \frac{5}{6} y - \frac{5}{6} \cdot 1 = \frac{3}{4} \cdot (y - 3) - 2$

Combine $y$ and $\frac{5}{6}$ .

$\frac{y}{3} - \frac{2}{3} - \frac{y \cdot 5}{6} - \frac{5}{6} \cdot 1 = \frac{3}{4} \cdot (y - 3) - 2$

Multiply $- 1$ by $1$ .

$\frac{y}{3} - \frac{2}{3} - \frac{y \cdot 5}{6} - \frac{5}{6} = \frac{3}{4} \cdot (y - 3) - 2$

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

$\frac{y}{3} - \frac{2}{3} - \frac{5 y}{6} - \frac{5}{6} = \frac{3}{4} \cdot (y - 3) - 2$

To write $\frac{y}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{3} \cdot \frac{2}{2} - \frac{5 y}{6} - \frac{2}{3} - \frac{5}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{y}{3}$ by $\frac{2}{2}$ .

$\frac{y \cdot 2}{3 \cdot 2} - \frac{5 y}{6} - \frac{2}{3} - \frac{5}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Multiply $3$ by $2$ .

$\frac{y \cdot 2}{6} - \frac{5 y}{6} - \frac{2}{3} - \frac{5}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{y \cdot 2 - 5 y}{6} - \frac{2}{3} - \frac{5}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Combine the numerators over the common denominator.

$\frac{y \cdot 2 - 5 y - 5}{6} + \frac{- 2}{3} = \frac{3}{4} \cdot (y - 3) - 2$

Move $2$ to the left of $y$ .

$\frac{2 y - 5 y - 5}{6} + \frac{- 2}{3} = \frac{3}{4} \cdot (y - 3) - 2$

Simplify by adding terms.

Subtract $5 y$ from $2 y$ .

$\frac{- 3 y - 5}{6} + \frac{- 2}{3} = \frac{3}{4} \cdot (y - 3) - 2$

Move the negative in front of the fraction.

$\frac{- 3 y - 5}{6} - \frac{2}{3} = \frac{3}{4} \cdot (y - 3) - 2$

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 3 y - 5}{6} - \frac{2}{3} \cdot \frac{2}{2} = \frac{3}{4} \cdot (y - 3) - 2$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$\frac{- 3 y - 5}{6} - \frac{2 \cdot 2}{3 \cdot 2} = \frac{3}{4} \cdot (y - 3) - 2$

Multiply $3$ by $2$ .

$\frac{- 3 y - 5}{6} - \frac{2 \cdot 2}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Combine the numerators over the common denominator.

$\frac{- 3 y - 5 - 2 \cdot 2}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$\frac{- 3 y - 5 - 4}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Subtract $4$ from $- 5$ .

$\frac{- 3 y - 9}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Factor $3$ out of $- 3 y - 9$ .

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

$\frac{3 (- y) - 9}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Factor $3$ out of $- 9$ .

$\frac{3 (- y) + 3 (- 3)}{6} = \frac{3}{4} \cdot (y - 3) - 2$

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

$\frac{3 (- y - 3)}{6} = \frac{3}{4} \cdot (y - 3) - 2$

Cancel the common factors.

Factor $3$ out of $6$ .

$\frac{3 (- y - 3)}{3 \cdot 2} = \frac{3}{4} \cdot (y - 3) - 2$

Cancel the common factor.

$\frac{3 (- y - 3)}{3 \cdot 2} = \frac{3}{4} \cdot (y - 3) - 2$

Rewrite the expression.

$\frac{- y - 3}{2} = \frac{3}{4} \cdot (y - 3) - 2$

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

$\frac{- (y) - 3}{2} = \frac{3}{4} \cdot (y - 3) - 2$

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (y) - 1 (3)}{2} = \frac{3}{4} \cdot (y - 3) - 2$

Factor $- 1$ out of $- (y) - 1 (3)$ .

$\frac{- (y + 3)}{2} = \frac{3}{4} \cdot (y - 3) - 2$

Simplify the expression.

Rewrite $- (y + 3)$ as $- 1 (y + 3)$ .

$\frac{- 1 (y + 3)}{2} = \frac{3}{4} \cdot (y - 3) - 2$

Move the negative in front of the fraction.

$- \frac{y + 3}{2} = \frac{3}{4} \cdot (y - 3) - 2$

Step 2

Simplify $\frac{3}{4} \cdot (y - 3) - 2$ .

Simplify each term.

Apply the distributive property.

$- \frac{y + 3}{2} = \frac{3}{4} y + \frac{3}{4} \cdot - 3 - 2$

Combine $\frac{3}{4}$ and $y$ .

$- \frac{y + 3}{2} = \frac{3 y}{4} + \frac{3}{4} \cdot - 3 - 2$

Multiply $\frac{3}{4} \cdot - 3$ .

Combine $\frac{3}{4}$ and $- 3$ .

$- \frac{y + 3}{2} = \frac{3 y}{4} + \frac{3 \cdot - 3}{4} - 2$

Multiply $3$ by $- 3$ .

$- \frac{y + 3}{2} = \frac{3 y}{4} + \frac{- 9}{4} - 2$

Move the negative in front of the fraction.

$- \frac{y + 3}{2} = \frac{3 y}{4} - \frac{9}{4} - 2$

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

$- \frac{y + 3}{2} = \frac{3 y}{4} - \frac{9}{4} - 2 \cdot \frac{4}{4}$

Combine $- 2$ and $\frac{4}{4}$ .

$- \frac{y + 3}{2} = \frac{3 y}{4} - \frac{9}{4} + \frac{- 2 \cdot 4}{4}$

Combine the numerators over the common denominator.

$- \frac{y + 3}{2} = \frac{3 y}{4} + \frac{- 9 - 2 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 2$ by $4$ .

$- \frac{y + 3}{2} = \frac{3 y}{4} + \frac{- 9 - 8}{4}$

Subtract $8$ from $- 9$ .

$- \frac{y + 3}{2} = \frac{3 y}{4} + \frac{- 17}{4}$

Move the negative in front of the fraction.

$- \frac{y + 3}{2} = \frac{3 y}{4} - \frac{17}{4}$

Step 3

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

Subtract $\frac{3 y}{4}$ from both sides of the equation.

$- \frac{y + 3}{2} - \frac{3 y}{4} = - \frac{17}{4}$

To write $- \frac{y + 3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{y + 3}{2} \cdot \frac{2}{2} - \frac{3 y}{4} = - \frac{17}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{y + 3}{2}$ by $\frac{2}{2}$ .

$- \frac{(y + 3) \cdot 2}{2 \cdot 2} - \frac{3 y}{4} = - \frac{17}{4}$

Multiply $2$ by $2$ .

$- \frac{(y + 3) \cdot 2}{4} - \frac{3 y}{4} = - \frac{17}{4}$

Combine the numerators over the common denominator.

$\frac{- (y + 3) \cdot 2 - 3 y}{4} = - \frac{17}{4}$

Simplify the numerator.

Apply the distributive property.

$\frac{(- y - 1 \cdot 3) \cdot 2 - 3 y}{4} = - \frac{17}{4}$

Multiply $- 1$ by $3$ .

$\frac{(- y - 3) \cdot 2 - 3 y}{4} = - \frac{17}{4}$

Apply the distributive property.

$\frac{- y \cdot 2 - 3 \cdot 2 - 3 y}{4} = - \frac{17}{4}$

Multiply $2$ by $- 1$ .

$\frac{- 2 y - 3 \cdot 2 - 3 y}{4} = - \frac{17}{4}$

Multiply $- 3$ by $2$ .

$\frac{- 2 y - 6 - 3 y}{4} = - \frac{17}{4}$

Subtract $3 y$ from $- 2 y$ .

$\frac{- 5 y - 6}{4} = - \frac{17}{4}$

Factor $- 1$ out of $- 5 y$ .

$\frac{- (5 y) - 6}{4} = - \frac{17}{4}$

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- (5 y) - 1 (6)}{4} = - \frac{17}{4}$

Factor $- 1$ out of $- (5 y) - 1 (6)$ .

$\frac{- (5 y + 6)}{4} = - \frac{17}{4}$

Rewrite $- (5 y + 6)$ as $- 1 (5 y + 6)$ .

$\frac{- 1 (5 y + 6)}{4} = - \frac{17}{4}$

Move the negative in front of the fraction.

$- \frac{5 y + 6}{4} = - \frac{17}{4}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- (5 y + 6) = - 17$

Step 5

Simplify $- (5 y + 6)$ .

Apply the distributive property.

$- (5 y) - 1 \cdot 6 = - 17$

Multiply.

Multiply $5$ by $- 1$ .

$- 5 y - 1 \cdot 6 = - 17$

Multiply $- 1$ by $6$ .

$- 5 y - 6 = - 17$

Step 6

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

Add $6$ to both sides of the equation.

$- 5 y = - 17 + 6$

Add $- 17$ and $6$ .

$- 5 y = - 11$

Step 7

Divide each term in $- 5 y = - 11$ by $- 5$ and simplify.

Divide each term in $- 5 y = - 11$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 11}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 11}{- 5}$

Divide $y$ by $1$ .

$y = \frac{- 11}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$y = \frac{11}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = \frac{11}{5}$

Decimal Form:

$y = 2.2$

Mixed Number Form:

$y = 2 \frac{1}{5}$

Solve for y 1/3(y-2)-5/6(y+1)=3/4(y-3)-2

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