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

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

Step 1

Simplify $- \frac{1}{3} y + \frac{3}{5} y + \frac{3}{4} y$ .

Simplify each term.

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

$- \frac{y}{3} + \frac{3}{5} y + \frac{3}{4} y = - \frac{2}{5}$

Combine $\frac{3}{5}$ and $y$ .

$- \frac{y}{3} + \frac{3 y}{5} + \frac{3}{4} y = - \frac{2}{5}$

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

$- \frac{y}{3} + \frac{3 y}{5} + \frac{3 y}{4} = - \frac{2}{5}$

Find the common denominator.

Multiply $\frac{y}{3}$ by $\frac{20}{20}$ .

$- (\frac{y}{3} \cdot \frac{20}{20}) + \frac{3 y}{5} + \frac{3 y}{4} = - \frac{2}{5}$

Multiply $\frac{y}{3}$ by $\frac{20}{20}$ .

$- \frac{y \cdot 20}{3 \cdot 20} + \frac{3 y}{5} + \frac{3 y}{4} = - \frac{2}{5}$

Multiply $\frac{3 y}{5}$ by $\frac{12}{12}$ .

$- \frac{y \cdot 20}{3 \cdot 20} + \frac{3 y}{5} \cdot \frac{12}{12} + \frac{3 y}{4} = - \frac{2}{5}$

Multiply $\frac{3 y}{5}$ by $\frac{12}{12}$ .

$- \frac{y \cdot 20}{3 \cdot 20} + \frac{3 y \cdot 12}{5 \cdot 12} + \frac{3 y}{4} = - \frac{2}{5}$

Multiply $\frac{3 y}{4}$ by $\frac{15}{15}$ .

$- \frac{y \cdot 20}{3 \cdot 20} + \frac{3 y \cdot 12}{5 \cdot 12} + \frac{3 y}{4} \cdot \frac{15}{15} = - \frac{2}{5}$

Multiply $\frac{3 y}{4}$ by $\frac{15}{15}$ .

$- \frac{y \cdot 20}{3 \cdot 20} + \frac{3 y \cdot 12}{5 \cdot 12} + \frac{3 y \cdot 15}{4 \cdot 15} = - \frac{2}{5}$

Multiply $3$ by $20$ .

$- \frac{y \cdot 20}{60} + \frac{3 y \cdot 12}{5 \cdot 12} + \frac{3 y \cdot 15}{4 \cdot 15} = - \frac{2}{5}$

Multiply $5$ by $12$ .

$- \frac{y \cdot 20}{60} + \frac{3 y \cdot 12}{60} + \frac{3 y \cdot 15}{4 \cdot 15} = - \frac{2}{5}$

Multiply $4$ by $15$ .

$- \frac{y \cdot 20}{60} + \frac{3 y \cdot 12}{60} + \frac{3 y \cdot 15}{60} = - \frac{2}{5}$

Combine the numerators over the common denominator.

$\frac{- y \cdot 20 + 3 y \cdot 12 + 3 y \cdot 15}{60} = - \frac{2}{5}$

Simplify each term.

Multiply $20$ by $- 1$ .

$\frac{- 20 y + 3 y \cdot 12 + 3 y \cdot 15}{60} = - \frac{2}{5}$

Multiply $12$ by $3$ .

$\frac{- 20 y + 36 y + 3 y \cdot 15}{60} = - \frac{2}{5}$

Multiply $15$ by $3$ .

$\frac{- 20 y + 36 y + 45 y}{60} = - \frac{2}{5}$

Simplify by adding terms.

Add $- 20 y$ and $36 y$ .

$\frac{16 y + 45 y}{60} = - \frac{2}{5}$

Add $16 y$ and $45 y$ .

$\frac{61 y}{60} = - \frac{2}{5}$

Step 2

Multiply both sides of the equation by $\frac{60}{61}$ .

$\frac{60}{61} \cdot \frac{61 y}{60} = \frac{60}{61} (- \frac{2}{5})$

Step 3

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{60}{61} \cdot \frac{61 y}{60}$ .

Cancel the common factor of $60$ .

Cancel the common factor.

$\frac{60}{61} \cdot \frac{61 y}{60} = \frac{60}{61} (- \frac{2}{5})$

Rewrite the expression.

$\frac{1}{61} (61 y) = \frac{60}{61} (- \frac{2}{5})$

Cancel the common factor of $61$ .

Factor $61$ out of $61 y$ .

$\frac{1}{61} (61 (y)) = \frac{60}{61} (- \frac{2}{5})$

Cancel the common factor.

$\frac{1}{61} (61 y) = \frac{60}{61} (- \frac{2}{5})$

Rewrite the expression.

$y = \frac{60}{61} (- \frac{2}{5})$

Simplify the right side.

Simplify $\frac{60}{61} (- \frac{2}{5})$ .

Cancel the common factor of $5$ .

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

$y = \frac{60}{61} \cdot \frac{- 2}{5}$

Factor $5$ out of $60$ .

$y = \frac{5 (12)}{61} \cdot \frac{- 2}{5}$

Cancel the common factor.

$y = \frac{5 \cdot 12}{61} \cdot \frac{- 2}{5}$

Rewrite the expression.

$y = \frac{12}{61} \cdot - 2$

Combine $\frac{12}{61}$ and $- 2$ .

$y = \frac{12 \cdot - 2}{61}$

Simplify the expression.

Multiply $12$ by $- 2$ .

$y = \frac{- 24}{61}$

Move the negative in front of the fraction.

$y = - \frac{24}{61}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{24}{61}$

Decimal Form:

$y = - 0.39344262 \dots$

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

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