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

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

Step 1

Simplify $\frac{1}{3} \cdot (n + 1) + 2$ .

Simplify each term.

Apply the distributive property.

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

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

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

Multiply $\frac{1}{3}$ by $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$\frac{n}{3} + \frac{1 + 6}{3} = \frac{1}{6} \cdot (3 n - 5)$

Add $1$ and $6$ .

$\frac{n}{3} + \frac{7}{3} = \frac{1}{6} \cdot (3 n - 5)$

Step 2

Simplify $\frac{1}{6} \cdot (3 n - 5)$ .

Apply the distributive property.

$\frac{n}{3} + \frac{7}{3} = \frac{1}{6} (3 n) + \frac{1}{6} \cdot - 5$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$\frac{n}{3} + \frac{7}{3} = \frac{1}{3 (2)} (3 n) + \frac{1}{6} \cdot - 5$

Factor $3$ out of $3 n$ .

$\frac{n}{3} + \frac{7}{3} = \frac{1}{3 (2)} (3 (n)) + \frac{1}{6} \cdot - 5$

Cancel the common factor.

$\frac{n}{3} + \frac{7}{3} = \frac{1}{3 \cdot 2} (3 n) + \frac{1}{6} \cdot - 5$

Rewrite the expression.

$\frac{n}{3} + \frac{7}{3} = \frac{1}{2} n + \frac{1}{6} \cdot - 5$

Combine $\frac{1}{2}$ and $n$ .

$\frac{n}{3} + \frac{7}{3} = \frac{n}{2} + \frac{1}{6} \cdot - 5$

Combine $\frac{1}{6}$ and $- 5$ .

$\frac{n}{3} + \frac{7}{3} = \frac{n}{2} + \frac{- 5}{6}$

Move the negative in front of the fraction.

$\frac{n}{3} + \frac{7}{3} = \frac{n}{2} - \frac{5}{6}$

Step 3

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

Subtract $\frac{n}{2}$ from both sides of the equation.

$\frac{n}{3} + \frac{7}{3} - \frac{n}{2} = - \frac{5}{6}$

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

$\frac{n}{3} \cdot \frac{2}{2} - \frac{n}{2} + \frac{7}{3} = - \frac{5}{6}$

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

$\frac{n}{3} \cdot \frac{2}{2} - \frac{n}{2} \cdot \frac{3}{3} + \frac{7}{3} = - \frac{5}{6}$

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

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

$\frac{n \cdot 2}{3 \cdot 2} - \frac{n}{2} \cdot \frac{3}{3} + \frac{7}{3} = - \frac{5}{6}$

Multiply $3$ by $2$ .

$\frac{n \cdot 2}{6} - \frac{n}{2} \cdot \frac{3}{3} + \frac{7}{3} = - \frac{5}{6}$

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

$\frac{n \cdot 2}{6} - \frac{n \cdot 3}{2 \cdot 3} + \frac{7}{3} = - \frac{5}{6}$

Multiply $2$ by $3$ .

$\frac{n \cdot 2}{6} - \frac{n \cdot 3}{6} + \frac{7}{3} = - \frac{5}{6}$

Combine the numerators over the common denominator.

$\frac{n \cdot 2 - n \cdot 3}{6} + \frac{7}{3} = - \frac{5}{6}$

Simplify each term.

Simplify the numerator.

Factor $n$ out of $n \cdot 2 - n \cdot 3$ .

Factor $n$ out of $n \cdot 2$ .

$\frac{n (2) - n \cdot 3}{6} + \frac{7}{3} = - \frac{5}{6}$

Factor $n$ out of $- n \cdot 3$ .

$\frac{n (2) + n (- 1 \cdot 3)}{6} + \frac{7}{3} = - \frac{5}{6}$

Factor $n$ out of $n (2) + n (- 1 \cdot 3)$ .

$\frac{n (2 - 1 \cdot 3)}{6} + \frac{7}{3} = - \frac{5}{6}$

Multiply $- 1$ by $3$ .

$\frac{n (2 - 3)}{6} + \frac{7}{3} = - \frac{5}{6}$

Subtract $3$ from $2$ .

$\frac{n \cdot - 1}{6} + \frac{7}{3} = - \frac{5}{6}$

Move $- 1$ to the left of $n$ .

$\frac{- 1 \cdot n}{6} + \frac{7}{3} = - \frac{5}{6}$

Move the negative in front of the fraction.

$- \frac{n}{6} + \frac{7}{3} = - \frac{5}{6}$

Step 4

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

Subtract $\frac{7}{3}$ from both sides of the equation.

$- \frac{n}{6} = - \frac{5}{6} - \frac{7}{3}$

To write $- \frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{n}{6} = - \frac{5}{6} - \frac{7}{3} \cdot \frac{2}{2}$

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

Multiply $\frac{7}{3}$ by $\frac{2}{2}$ .

$- \frac{n}{6} = - \frac{5}{6} - \frac{7 \cdot 2}{3 \cdot 2}$

Multiply $3$ by $2$ .

$- \frac{n}{6} = - \frac{5}{6} - \frac{7 \cdot 2}{6}$

Combine the numerators over the common denominator.

$- \frac{n}{6} = \frac{- 5 - 7 \cdot 2}{6}$

Simplify the numerator.

Multiply $- 7$ by $2$ .

$- \frac{n}{6} = \frac{- 5 - 14}{6}$

Subtract $14$ from $- 5$ .

$- \frac{n}{6} = \frac{- 19}{6}$

Move the negative in front of the fraction.

$- \frac{n}{6} = - \frac{19}{6}$

Step 5

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

$- n = - 19$

Step 6

Divide each term in $- n = - 19$ by $- 1$ and simplify.

Divide each term in $- n = - 19$ by $- 1$ .

$\frac{- n}{- 1} = \frac{- 19}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{n}{1} = \frac{- 19}{- 1}$

Divide $n$ by $1$ .

$n = \frac{- 19}{- 1}$

Simplify the right side.

Divide $- 19$ by $- 1$ .

$n = 19$

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

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