Solve for a (a+13)/10-(2a)/5=(3-a)/15+a/2

$\frac{a + 13}{10} - \frac{2 a}{5} = \frac{3 - a}{15} + \frac{a}{2}$

Simplify $\frac{a + 13}{10} - \frac{2 a}{5}$ .

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

$\frac{a + 13}{10} - \frac{2 a}{5} \cdot \frac{2}{2} = \frac{3 - a}{15} + \frac{a}{2}$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 a}{5}$ by $\frac{2}{2}$ .

$\frac{a + 13}{10} - \frac{2 a \cdot 2}{5 \cdot 2} = \frac{3 - a}{15} + \frac{a}{2}$

Multiply $5$ by $2$ .

$\frac{a + 13}{10} - \frac{2 a \cdot 2}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Combine the numerators over the common denominator.

$\frac{a + 13 - 2 a \cdot 2}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Simplify the numerator.

Multiply $2$ by $- 2$ .

$\frac{a + 13 - 4 a}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Subtract $4 a$ from $a$ .

$\frac{- 3 a + 13}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Simplify with factoring out.

Factor $- 1$ out of $- 3 a$ .

$\frac{- (3 a) + 13}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (3 a) - 1 (- 13)}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Factor $- 1$ out of $- (3 a) - 1 (- 13)$ .

$\frac{- (3 a - 13)}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Simplify the expression.

Rewrite $- (3 a - 13)$ as $- 1 (3 a - 13)$ .

$\frac{- 1 (3 a - 13)}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Move the negative in front of the fraction.

$- \frac{3 a - 13}{10} = \frac{3 - a}{15} + \frac{a}{2}$

Simplify $\frac{3 - a}{15} + \frac{a}{2}$ .

To write $\frac{3 - a}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 a - 13}{10} = \frac{3 - a}{15} \cdot \frac{2}{2} + \frac{a}{2}$

To write $\frac{a}{2}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$- \frac{3 a - 13}{10} = \frac{3 - a}{15} \cdot \frac{2}{2} + \frac{a}{2} \cdot \frac{15}{15}$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 - a}{15}$ by $\frac{2}{2}$ .

$- \frac{3 a - 13}{10} = \frac{(3 - a) \cdot 2}{15 \cdot 2} + \frac{a}{2} \cdot \frac{15}{15}$

Multiply $15$ by $2$ .

$- \frac{3 a - 13}{10} = \frac{(3 - a) \cdot 2}{30} + \frac{a}{2} \cdot \frac{15}{15}$

Multiply $\frac{a}{2}$ by $\frac{15}{15}$ .

$- \frac{3 a - 13}{10} = \frac{(3 - a) \cdot 2}{30} + \frac{a \cdot 15}{2 \cdot 15}$

Multiply $2$ by $15$ .

$- \frac{3 a - 13}{10} = \frac{(3 - a) \cdot 2}{30} + \frac{a \cdot 15}{30}$

Combine the numerators over the common denominator.

$- \frac{3 a - 13}{10} = \frac{(3 - a) \cdot 2 + a \cdot 15}{30}$

Simplify the numerator.

Apply the distributive property.

$- \frac{3 a - 13}{10} = \frac{3 \cdot 2 - a \cdot 2 + a \cdot 15}{30}$

Multiply $3$ by $2$ .

$- \frac{3 a - 13}{10} = \frac{6 - a \cdot 2 + a \cdot 15}{30}$

Multiply $2$ by $- 1$ .

$- \frac{3 a - 13}{10} = \frac{6 - 2 a + a \cdot 15}{30}$

Move $15$ to the left of $a$ .

$- \frac{3 a - 13}{10} = \frac{6 - 2 a + 15 \cdot a}{30}$

Add $- 2 a$ and $15 a$ .

$- \frac{3 a - 13}{10} = \frac{6 + 13 a}{30}$

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

Subtract $\frac{6 + 13 a}{30}$ from both sides of the equation.

$- \frac{3 a - 13}{10} - \frac{6 + 13 a}{30} = 0$

To write $- \frac{3 a - 13}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{3 a - 13}{10} \cdot \frac{3}{3} - \frac{6 + 13 a}{30} = 0$

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 a - 13}{10}$ by $\frac{3}{3}$ .

$- \frac{(3 a - 13) \cdot 3}{10 \cdot 3} - \frac{6 + 13 a}{30} = 0$

Multiply $10$ by $3$ .

$- \frac{(3 a - 13) \cdot 3}{30} - \frac{6 + 13 a}{30} = 0$

Combine the numerators over the common denominator.

$\frac{- (3 a - 13) \cdot 3 - (6 + 13 a)}{30} = 0$

Simplify the numerator.

Factor $- 1$ out of $- (3 a - 13) \cdot 3 - (6 + 13 a)$ .

Reorder the expression.

Move $3 a - 13$ .

$\frac{- 1 \cdot 3 (3 a - 13) - (6 + 13 a)}{30} = 0$

Reorder $6$ and $13 a$ .

$\frac{- 1 \cdot 3 (3 a - 13) - (13 a + 6)}{30} = 0$

Factor $- 1$ out of $- 1 \cdot 3 (3 a - 13)$ .

$\frac{- (3 (3 a - 13)) - (13 a + 6)}{30} = 0$

Factor $- 1$ out of $- (3 (3 a - 13)) - (13 a + 6)$ .

$\frac{- (3 (3 a - 13) + 13 a + 6)}{30} = 0$

Apply the distributive property.

$\frac{- (3 (3 a) + 3 \cdot - 13 + 13 a + 6)}{30} = 0$

Multiply $3$ by $3$ .

$\frac{- (9 a + 3 \cdot - 13 + 13 a + 6)}{30} = 0$

Multiply $3$ by $- 13$ .

$\frac{- (9 a - 39 + 13 a + 6)}{30} = 0$

Add $9 a$ and $13 a$ .

$\frac{- (22 a - 39 + 6)}{30} = 0$

Add $- 39$ and $6$ .

$\frac{- (22 a - 33)}{30} = 0$

Factor $11$ out of $22 a - 33$ .

Factor $11$ out of $22 a$ .

$\frac{- (11 (2 a) - 33)}{30} = 0$

Factor $11$ out of $- 33$ .

$\frac{- (11 (2 a) + 11 (- 3))}{30} = 0$

Factor $11$ out of $11 (2 a) + 11 (- 3)$ .

$\frac{- (11 (2 a - 3))}{30} = 0$

$\frac{- 1 \cdot 11 (2 a - 3)}{30} = 0$

Multiply $- 1$ by $11$ .

$\frac{- 11 (2 a - 3)}{30} = 0$

Move the negative in front of the fraction.

$- \frac{11 (2 a - 3)}{30} = 0$

Set the numerator equal to zero.

$11 (2 a - 3) = 0$

Solve the equation for $a$ .

Divide each term in $11 (2 a - 3) = 0$ by $11$ and simplify.

Divide each term in $11 (2 a - 3) = 0$ by $11$ .

$\frac{11 (2 a - 3)}{11} = \frac{0}{11}$

Simplify the left side.

Cancel the common factor of $11$ .

Cancel the common factor.

$\frac{11 (2 a - 3)}{11} = \frac{0}{11}$

Divide $2 a - 3$ by $1$ .

$2 a - 3 = \frac{0}{11}$

Simplify the right side.

Divide $0$ by $11$ .

$2 a - 3 = 0$

Add $3$ to both sides of the equation.

$2 a = 3$

Divide each term in $2 a = 3$ by $2$ and simplify.

Divide each term in $2 a = 3$ by $2$ .

$\frac{2 a}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{3}{2}$

Divide $a$ by $1$ .

$a = \frac{3}{2}$

The result can be shown in multiple forms.

Exact Form:

$a = \frac{3}{2}$

Decimal Form:

$a = 1.5$

Mixed Number Form:

$a = 1 \frac{1}{2}$

Solve for a (a+13)/10-(2a)/5=(3-a)/15+a/2

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