Evaluate (n-5)/6+(n+3)/7=11/21

$\frac{n - 5}{6} + \frac{n + 3}{7} = \frac{11}{21}$

Step 1

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

To write $\frac{n - 5}{6}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{n - 5}{6} \cdot \frac{7}{7} + \frac{n + 3}{7} = \frac{11}{21}$

To write $\frac{n + 3}{7}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{n - 5}{6} \cdot \frac{7}{7} + \frac{n + 3}{7} \cdot \frac{6}{6} = \frac{11}{21}$

Write each expression with a common denominator of $42$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{n - 5}{6}$ by $\frac{7}{7}$ .

$\frac{(n - 5) \cdot 7}{6 \cdot 7} + \frac{n + 3}{7} \cdot \frac{6}{6} = \frac{11}{21}$

Multiply $6$ by $7$ .

$\frac{(n - 5) \cdot 7}{42} + \frac{n + 3}{7} \cdot \frac{6}{6} = \frac{11}{21}$

Multiply $\frac{n + 3}{7}$ by $\frac{6}{6}$ .

$\frac{(n - 5) \cdot 7}{42} + \frac{(n + 3) \cdot 6}{7 \cdot 6} = \frac{11}{21}$

Multiply $7$ by $6$ .

$\frac{(n - 5) \cdot 7}{42} + \frac{(n + 3) \cdot 6}{42} = \frac{11}{21}$

Combine the numerators over the common denominator.

$\frac{(n - 5) \cdot 7 + (n + 3) \cdot 6}{42} = \frac{11}{21}$

Simplify the numerator.

Apply the distributive property.

$\frac{n \cdot 7 - 5 \cdot 7 + (n + 3) \cdot 6}{42} = \frac{11}{21}$

Move $7$ to the left of $n$ .

$\frac{7 \cdot n - 5 \cdot 7 + (n + 3) \cdot 6}{42} = \frac{11}{21}$

Multiply $- 5$ by $7$ .

$\frac{7 \cdot n - 35 + (n + 3) \cdot 6}{42} = \frac{11}{21}$

Apply the distributive property.

$\frac{7 n - 35 + n \cdot 6 + 3 \cdot 6}{42} = \frac{11}{21}$

Move $6$ to the left of $n$ .

$\frac{7 n - 35 + 6 \cdot n + 3 \cdot 6}{42} = \frac{11}{21}$

Multiply $3$ by $6$ .

$\frac{7 n - 35 + 6 \cdot n + 18}{42} = \frac{11}{21}$

Add $7 n$ and $6 n$ .

$\frac{13 n - 35 + 18}{42} = \frac{11}{21}$

Add $- 35$ and $18$ .

$\frac{13 n - 17}{42} = \frac{11}{21}$

Step 2

Multiply both sides by $42$ .

$\frac{13 n - 17}{42} \cdot 42 = \frac{11}{21} \cdot 42$

Step 3

Simplify.

Simplify the left side.

Cancel the common factor of $42$ .

Cancel the common factor.

$\frac{13 n - 17}{42} \cdot 42 = \frac{11}{21} \cdot 42$

Rewrite the expression.

$13 n - 17 = \frac{11}{21} \cdot 42$

Simplify the right side.

Simplify $\frac{11}{21} \cdot 42$ .

Cancel the common factor of $21$ .

Factor $21$ out of $42$ .

$13 n - 17 = \frac{11}{21} \cdot (21 (2))$

Cancel the common factor.

$13 n - 17 = \frac{11}{21} \cdot (21 \cdot 2)$

Rewrite the expression.

$13 n - 17 = 11 \cdot 2$

Multiply $11$ by $2$ .

$13 n - 17 = 22$

Step 4

Solve for $n$ .

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

Add $17$ to both sides of the equation.

$13 n = 22 + 17$

Add $22$ and $17$ .

$13 n = 39$

Divide each term in $13 n = 39$ by $13$ and simplify.

Divide each term in $13 n = 39$ by $13$ .

$\frac{13 n}{13} = \frac{39}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 n}{13} = \frac{39}{13}$

Divide $n$ by $1$ .

$n = \frac{39}{13}$

Simplify the right side.

Divide $39$ by $13$ .

$n = 3$

Evaluate (n-5)/6+(n+3)/7=11/21

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