Solve for a (3a-5)/12+(a+4)/2=(5a-7)/14+a/2

$\frac{3 a - 5}{12} + \frac{a + 4}{2} = \frac{5 a - 7}{14} + \frac{a}{2}$

Step 1

Simplify $\frac{3 a - 5}{12} + \frac{a + 4}{2}$ .

To write $\frac{a + 4}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{3 a - 5}{12} + \frac{a + 4}{2} \cdot \frac{6}{6} = \frac{5 a - 7}{14} + \frac{a}{2}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a + 4}{2}$ by $\frac{6}{6}$ .

$\frac{3 a - 5}{12} + \frac{(a + 4) \cdot 6}{2 \cdot 6} = \frac{5 a - 7}{14} + \frac{a}{2}$

Multiply $2$ by $6$ .

$\frac{3 a - 5}{12} + \frac{(a + 4) \cdot 6}{12} = \frac{5 a - 7}{14} + \frac{a}{2}$

Combine the numerators over the common denominator.

$\frac{3 a - 5 + (a + 4) \cdot 6}{12} = \frac{5 a - 7}{14} + \frac{a}{2}$

Simplify the numerator.

Apply the distributive property.

$\frac{3 a - 5 + a \cdot 6 + 4 \cdot 6}{12} = \frac{5 a - 7}{14} + \frac{a}{2}$

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

$\frac{3 a - 5 + 6 \cdot a + 4 \cdot 6}{12} = \frac{5 a - 7}{14} + \frac{a}{2}$

Multiply $4$ by $6$ .

$\frac{3 a - 5 + 6 \cdot a + 24}{12} = \frac{5 a - 7}{14} + \frac{a}{2}$

Add $3 a$ and $6 a$ .

$\frac{9 a - 5 + 24}{12} = \frac{5 a - 7}{14} + \frac{a}{2}$

Add $- 5$ and $24$ .

$\frac{9 a + 19}{12} = \frac{5 a - 7}{14} + \frac{a}{2}$

Step 2

Simplify $\frac{5 a - 7}{14} + \frac{a}{2}$ .

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

$\frac{9 a + 19}{12} = \frac{5 a - 7}{14} + \frac{a}{2} \cdot \frac{7}{7}$

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{9 a + 19}{12} = \frac{5 a - 7}{14} + \frac{a \cdot 7}{2 \cdot 7}$

Multiply $2$ by $7$ .

$\frac{9 a + 19}{12} = \frac{5 a - 7}{14} + \frac{a \cdot 7}{14}$

Combine the numerators over the common denominator.

$\frac{9 a + 19}{12} = \frac{5 a - 7 + a \cdot 7}{14}$

Simplify the numerator.

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

$\frac{9 a + 19}{12} = \frac{5 a - 7 + 7 \cdot a}{14}$

Add $5 a$ and $7 a$ .

$\frac{9 a + 19}{12} = \frac{12 a - 7}{14}$

Step 3

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

Subtract $\frac{12 a - 7}{14}$ from both sides of the equation.

$\frac{9 a + 19}{12} - \frac{12 a - 7}{14} = 0$

To write $\frac{9 a + 19}{12}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{9 a + 19}{12} \cdot \frac{7}{7} - \frac{12 a - 7}{14} = 0$

To write $- \frac{12 a - 7}{14}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{9 a + 19}{12} \cdot \frac{7}{7} - \frac{12 a - 7}{14} \cdot \frac{6}{6} = 0$

Write each expression with a common denominator of $84$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{9 a + 19}{12}$ by $\frac{7}{7}$ .

$\frac{(9 a + 19) \cdot 7}{12 \cdot 7} - \frac{12 a - 7}{14} \cdot \frac{6}{6} = 0$

Multiply $12$ by $7$ .

$\frac{(9 a + 19) \cdot 7}{84} - \frac{12 a - 7}{14} \cdot \frac{6}{6} = 0$

Multiply $\frac{12 a - 7}{14}$ by $\frac{6}{6}$ .

$\frac{(9 a + 19) \cdot 7}{84} - \frac{(12 a - 7) \cdot 6}{14 \cdot 6} = 0$

Multiply $14$ by $6$ .

$\frac{(9 a + 19) \cdot 7}{84} - \frac{(12 a - 7) \cdot 6}{84} = 0$

Combine the numerators over the common denominator.

$\frac{(9 a + 19) \cdot 7 - (12 a - 7) \cdot 6}{84} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{9 a \cdot 7 + 19 \cdot 7 - (12 a - 7) \cdot 6}{84} = 0$

Multiply $7$ by $9$ .

$\frac{63 a + 19 \cdot 7 - (12 a - 7) \cdot 6}{84} = 0$

Multiply $19$ by $7$ .

$\frac{63 a + 133 - (12 a - 7) \cdot 6}{84} = 0$

Apply the distributive property.

$\frac{63 a + 133 + (- (12 a) - - 7) \cdot 6}{84} = 0$

Multiply $12$ by $- 1$ .

$\frac{63 a + 133 + (- 12 a - - 7) \cdot 6}{84} = 0$

Multiply $- 1$ by $- 7$ .

$\frac{63 a + 133 + (- 12 a + 7) \cdot 6}{84} = 0$

Apply the distributive property.

$\frac{63 a + 133 - 12 a \cdot 6 + 7 \cdot 6}{84} = 0$

Multiply $6$ by $- 12$ .

$\frac{63 a + 133 - 72 a + 7 \cdot 6}{84} = 0$

Multiply $7$ by $6$ .

$\frac{63 a + 133 - 72 a + 42}{84} = 0$

Subtract $72 a$ from $63 a$ .

$\frac{- 9 a + 133 + 42}{84} = 0$

Add $133$ and $42$ .

$\frac{- 9 a + 175}{84} = 0$

Factor $- 1$ out of $- 9 a$ .

$\frac{- (9 a) + 175}{84} = 0$

Rewrite $175$ as $- 1 (- 175)$ .

$\frac{- (9 a) - 1 (- 175)}{84} = 0$

Factor $- 1$ out of $- (9 a) - 1 (- 175)$ .

$\frac{- (9 a - 175)}{84} = 0$

Rewrite $- (9 a - 175)$ as $- 1 (9 a - 175)$ .

$\frac{- 1 (9 a - 175)}{84} = 0$

Move the negative in front of the fraction.

$- \frac{9 a - 175}{84} = 0$

Step 4

Set the numerator equal to zero.

$9 a - 175 = 0$

Step 5

Solve the equation for $a$ .

Add $175$ to both sides of the equation.

$9 a = 175$

Divide each term in $9 a = 175$ by $9$ and simplify.

Divide each term in $9 a = 175$ by $9$ .

$\frac{9 a}{9} = \frac{175}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 a}{9} = \frac{175}{9}$

Divide $a$ by $1$ .

$a = \frac{175}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{175}{9}$

Decimal Form:

$a = 19.\overline{4}$

Mixed Number Form:

$a = 19 \frac{4}{9}$

Solve for a (3a-5)/12+(a+4)/2=(5a-7)/14+a/2

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