Basic Math Examples

$\frac{4 n - 7}{6} + \frac{1 - 3 n}{2} = \frac{2 - 5 n}{7}$

Step 1

Multiply both sides by $7$ .

$(\frac{4 n - 7}{6} + \frac{1 - 3 n}{2}) \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{4 n - 7}{6} + \frac{1 - 3 n}{2}) \cdot 7$ .

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Step 2.1.1.1

To write $\frac{1 - 3 n}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{4 n - 7}{6} + \frac{1 - 3 n}{2} \cdot \frac{3}{3}) \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.2

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

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Step 2.1.1.2.1

Multiply $\frac{1 - 3 n}{2}$ by $\frac{3}{3}$ .

$(\frac{4 n - 7}{6} + \frac{(1 - 3 n) \cdot 3}{2 \cdot 3}) \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.2.2

Multiply $2$ by $3$ .

$(\frac{4 n - 7}{6} + \frac{(1 - 3 n) \cdot 3}{6}) \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.3

Combine the numerators over the common denominator.

$\frac{4 n - 7 + (1 - 3 n) \cdot 3}{6} \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.4

Simplify the numerator.

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Step 2.1.1.4.1

Apply the distributive property.

$\frac{4 n - 7 + 1 \cdot 3 - 3 n \cdot 3}{6} \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.4.2

Multiply $3$ by $1$ .

$\frac{4 n - 7 + 3 - 3 n \cdot 3}{6} \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.4.3

Multiply $3$ by $- 3$ .

$\frac{4 n - 7 + 3 - 9 n}{6} \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.4.4

Subtract $9 n$ from $4 n$ .

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

Step 2.1.1.4.5

Add $- 7$ and $3$ .

$\frac{- 5 n - 4}{6} \cdot 7 = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.5

Simplify terms.

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Step 2.1.1.5.1

Combine $\frac{- 5 n - 4}{6}$ and $7$ .

$\frac{(- 5 n - 4) \cdot 7}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.5.2

Move $7$ to the left of $- 5 n - 4$ .

$\frac{7 \cdot (- 5 n - 4)}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.5.3

Factor $- 1$ out of $- 5 n$ .

$\frac{7 (- (5 n) - 4)}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.5.4

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{7 (- (5 n) - 1 (4))}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.5.5

Factor $- 1$ out of $- (5 n) - 1 (4)$ .

$\frac{7 (- (5 n + 4))}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.5.6

Simplify the expression.

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Step 2.1.1.5.6.1

Rewrite $- (5 n + 4)$ as $- 1 (5 n + 4)$ .

$\frac{7 (- 1 (5 n + 4))}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.1.1.5.6.2

Move the negative in front of the fraction.

$- \frac{7 (5 n + 4)}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{2 - 5 n}{7} \cdot 7$ .

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Step 2.2.1.1

Cancel the common factor of $7$ .

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Step 2.2.1.1.1

Cancel the common factor.

$- \frac{7 (5 n + 4)}{6} = \frac{2 - 5 n}{7} \cdot 7$

Step 2.2.1.1.2

Rewrite the expression.

$- \frac{7 (5 n + 4)}{6} = 2 - 5 n$

Step 2.2.1.2

Reorder $2$ and $- 5 n$ .

$- \frac{7 (5 n + 4)}{6} = - 5 n + 2$

Step 3

Solve for $n$ .

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Step 3.1

Multiply both sides by $6$ .

$- \frac{7 (5 n + 4)}{6} \cdot 6 = (- 5 n + 2) \cdot 6$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- \frac{7 (5 n + 4)}{6} \cdot 6$ .

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Step 3.2.1.1.1

Cancel the common factor of $6$ .

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Step 3.2.1.1.1.1

Move the leading negative in $- \frac{7 (5 n + 4)}{6}$ into the numerator.

$\frac{- 7 (5 n + 4)}{6} \cdot 6 = (- 5 n + 2) \cdot 6$

Step 3.2.1.1.1.2

Cancel the common factor.

$\frac{- 7 (5 n + 4)}{6} \cdot 6 = (- 5 n + 2) \cdot 6$

Step 3.2.1.1.1.3

Rewrite the expression.

$- 7 (5 n + 4) = (- 5 n + 2) \cdot 6$

Step 3.2.1.1.2

Apply the distributive property.

$- 7 (5 n) - 7 \cdot 4 = (- 5 n + 2) \cdot 6$

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $5$ by $- 7$ .

$- 35 n - 7 \cdot 4 = (- 5 n + 2) \cdot 6$

Step 3.2.1.1.3.2

Multiply $- 7$ by $4$ .

$- 35 n - 28 = (- 5 n + 2) \cdot 6$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(- 5 n + 2) \cdot 6$ .

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Step 3.2.2.1.1

Apply the distributive property.

$- 35 n - 28 = - 5 n \cdot 6 + 2 \cdot 6$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $6$ by $- 5$ .

$- 35 n - 28 = - 30 n + 2 \cdot 6$

Step 3.2.2.1.2.2

Multiply $2$ by $6$ .

$- 35 n - 28 = - 30 n + 12$

Step 3.3

Solve for $n$ .

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Step 3.3.1

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

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Step 3.3.1.1

Add $30 n$ to both sides of the equation.

$- 35 n - 28 + 30 n = 12$

Step 3.3.1.2

Add $- 35 n$ and $30 n$ .

$- 5 n - 28 = 12$

Step 3.3.2

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

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Step 3.3.2.1

Add $28$ to both sides of the equation.

$- 5 n = 12 + 28$

Step 3.3.2.2

Add $12$ and $28$ .

$- 5 n = 40$

Step 3.3.3

Divide each term in $- 5 n = 40$ by $- 5$ and simplify.

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Step 3.3.3.1

Divide each term in $- 5 n = 40$ by $- 5$ .

$\frac{- 5 n}{- 5} = \frac{40}{- 5}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 5 n}{- 5} = \frac{40}{- 5}$

Step 3.3.3.2.1.2

Divide $n$ by $1$ .

$n = \frac{40}{- 5}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $40$ by $- 5$ .

$n = - 8$