Solve for a (3a+41)/2-(a-3)/5=(9-2a)/6

$\frac{3 a + 41}{2} - \frac{a - 3}{5} = \frac{9 - 2 a}{6}$

Step 1

Multiply both sides by $6$ .

$(\frac{3 a + 41}{2} - \frac{a - 3}{5}) \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Step 2

Simplify.

Simplify the left side.

Simplify $(\frac{3 a + 41}{2} - \frac{a - 3}{5}) \cdot 6$ .

To write $\frac{3 a + 41}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$(\frac{3 a + 41}{2} \cdot \frac{5}{5} - \frac{a - 3}{5}) \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

To write $- \frac{a - 3}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(\frac{3 a + 41}{2} \cdot \frac{5}{5} - \frac{a - 3}{5} \cdot \frac{2}{2}) \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

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

Multiply $\frac{3 a + 41}{2}$ by $\frac{5}{5}$ .

$(\frac{(3 a + 41) \cdot 5}{2 \cdot 5} - \frac{a - 3}{5} \cdot \frac{2}{2}) \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $2$ by $5$ .

$(\frac{(3 a + 41) \cdot 5}{10} - \frac{a - 3}{5} \cdot \frac{2}{2}) \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $\frac{a - 3}{5}$ by $\frac{2}{2}$ .

$(\frac{(3 a + 41) \cdot 5}{10} - \frac{(a - 3) \cdot 2}{5 \cdot 2}) \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $5$ by $2$ .

$(\frac{(3 a + 41) \cdot 5}{10} - \frac{(a - 3) \cdot 2}{10}) \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Combine the numerators over the common denominator.

$\frac{(3 a + 41) \cdot 5 - (a - 3) \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Simplify the numerator.

Apply the distributive property.

$\frac{3 a \cdot 5 + 41 \cdot 5 - (a - 3) \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $5$ by $3$ .

$\frac{15 a + 41 \cdot 5 - (a - 3) \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $41$ by $5$ .

$\frac{15 a + 205 - (a - 3) \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Apply the distributive property.

$\frac{15 a + 205 + (- a - - 3) \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $- 1$ by $- 3$ .

$\frac{15 a + 205 + (- a + 3) \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Apply the distributive property.

$\frac{15 a + 205 - a \cdot 2 + 3 \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $2$ by $- 1$ .

$\frac{15 a + 205 - 2 a + 3 \cdot 2}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Multiply $3$ by $2$ .

$\frac{15 a + 205 - 2 a + 6}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Subtract $2 a$ from $15 a$ .

$\frac{13 a + 205 + 6}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Add $205$ and $6$ .

$\frac{13 a + 211}{10} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Simplify terms.

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$\frac{13 a + 211}{2 (5)} \cdot 6 = \frac{9 - 2 a}{6} \cdot 6$

Factor $2$ out of $6$ .

$\frac{13 a + 211}{2 \cdot 5} \cdot (2 \cdot 3) = \frac{9 - 2 a}{6} \cdot 6$

Cancel the common factor.

$\frac{13 a + 211}{2 \cdot 5} \cdot (2 \cdot 3) = \frac{9 - 2 a}{6} \cdot 6$

Rewrite the expression.

$\frac{13 a + 211}{5} \cdot 3 = \frac{9 - 2 a}{6} \cdot 6$

Combine $\frac{13 a + 211}{5}$ and $3$ .

$\frac{(13 a + 211) \cdot 3}{5} = \frac{9 - 2 a}{6} \cdot 6$

Move $3$ to the left of $13 a + 211$ .

$\frac{3 (13 a + 211)}{5} = \frac{9 - 2 a}{6} \cdot 6$

Simplify the right side.

Simplify $\frac{9 - 2 a}{6} \cdot 6$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{3 (13 a + 211)}{5} = \frac{9 - 2 a}{6} \cdot 6$

Rewrite the expression.

$\frac{3 (13 a + 211)}{5} = 9 - 2 a$

Reorder $9$ and $- 2 a$ .

$\frac{3 (13 a + 211)}{5} = - 2 a + 9$

Step 3

Solve for $a$ .

Multiply both sides by $5$ .

$\frac{3 (13 a + 211)}{5} \cdot 5 = (- 2 a + 9) \cdot 5$

Simplify.

Simplify the left side.

Simplify $\frac{3 (13 a + 211)}{5} \cdot 5$ .

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{3 (13 a + 211)}{5} \cdot 5 = (- 2 a + 9) \cdot 5$

Rewrite the expression.

$3 (13 a + 211) = (- 2 a + 9) \cdot 5$

Apply the distributive property.

$3 (13 a) + 3 \cdot 211 = (- 2 a + 9) \cdot 5$

Multiply.

Multiply $13$ by $3$ .

$39 a + 3 \cdot 211 = (- 2 a + 9) \cdot 5$

Multiply $3$ by $211$ .

$39 a + 633 = (- 2 a + 9) \cdot 5$

Simplify the right side.

Simplify $(- 2 a + 9) \cdot 5$ .

Apply the distributive property.

$39 a + 633 = - 2 a \cdot 5 + 9 \cdot 5$

Multiply.

Multiply $5$ by $- 2$ .

$39 a + 633 = - 10 a + 9 \cdot 5$

Multiply $9$ by $5$ .

$39 a + 633 = - 10 a + 45$

Solve for $a$ .

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

Add $10 a$ to both sides of the equation.

$39 a + 633 + 10 a = 45$

Add $39 a$ and $10 a$ .

$49 a + 633 = 45$

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

Subtract $633$ from both sides of the equation.

$49 a = 45 - 633$

Subtract $633$ from $45$ .

$49 a = - 588$

Divide each term in $49 a = - 588$ by $49$ and simplify.

Divide each term in $49 a = - 588$ by $49$ .

$\frac{49 a}{49} = \frac{- 588}{49}$

Simplify the left side.

Cancel the common factor of $49$ .

Cancel the common factor.

$\frac{49 a}{49} = \frac{- 588}{49}$

Divide $a$ by $1$ .

$a = \frac{- 588}{49}$

Simplify the right side.

Divide $- 588$ by $49$ .

$a = - 12$

Solve for a (3a+41)/2-(a-3)/5=(9-2a)/6

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