Solve for x (x-5)/6+(3x+7)/12=2x-5

$\frac{x - 5}{6} + \frac{3 x + 7}{12} = 2 x - 5$

Step 1

Simplify $\frac{x - 5}{6} + \frac{3 x + 7}{12}$ .

To write $\frac{x - 5}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x - 5}{6} \cdot \frac{2}{2} + \frac{3 x + 7}{12} = 2 x - 5$

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

Multiply $\frac{x - 5}{6}$ by $\frac{2}{2}$ .

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

Multiply $6$ by $2$ .

$\frac{(x - 5) \cdot 2}{12} + \frac{3 x + 7}{12} = 2 x - 5$

Combine the numerators over the common denominator.

$\frac{(x - 5) \cdot 2 + 3 x + 7}{12} = 2 x - 5$

Simplify the numerator.

Apply the distributive property.

$\frac{x \cdot 2 - 5 \cdot 2 + 3 x + 7}{12} = 2 x - 5$

Move $2$ to the left of $x$ .

$\frac{2 \cdot x - 5 \cdot 2 + 3 x + 7}{12} = 2 x - 5$

Multiply $- 5$ by $2$ .

$\frac{2 \cdot x - 10 + 3 x + 7}{12} = 2 x - 5$

Add $2 x$ and $3 x$ .

$\frac{5 x - 10 + 7}{12} = 2 x - 5$

Add $- 10$ and $7$ .

$\frac{5 x - 3}{12} = 2 x - 5$

Step 2

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

Subtract $2 x$ from both sides of the equation.

$\frac{5 x - 3}{12} - 2 x = - 5$

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{5 x - 3}{12} - 2 x \cdot \frac{12}{12} = - 5$

Combine $- 2 x$ and $\frac{12}{12}$ .

$\frac{5 x - 3}{12} + \frac{- 2 x \cdot 12}{12} = - 5$

Combine the numerators over the common denominator.

$\frac{5 x - 3 - 2 x \cdot 12}{12} = - 5$

Simplify the numerator.

Multiply $12$ by $- 2$ .

$\frac{5 x - 3 - 24 x}{12} = - 5$

Subtract $24 x$ from $5 x$ .

$\frac{- 19 x - 3}{12} = - 5$

Factor $- 1$ out of $- 19 x$ .

$\frac{- (19 x) - 3}{12} = - 5$

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (19 x) - 1 (3)}{12} = - 5$

Factor $- 1$ out of $- (19 x) - 1 (3)$ .

$\frac{- (19 x + 3)}{12} = - 5$

Rewrite $- (19 x + 3)$ as $- 1 (19 x + 3)$ .

$\frac{- 1 (19 x + 3)}{12} = - 5$

Move the negative in front of the fraction.

$- \frac{19 x + 3}{12} = - 5$

Step 3

Multiply both sides of the equation by $- 12$ .

$- 12 (- \frac{19 x + 3}{12}) = - 12 \cdot - 5$

Step 4

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 12 (- \frac{19 x + 3}{12})$ .

Cancel the common factor of $12$ .

Move the leading negative in $- \frac{19 x + 3}{12}$ into the numerator.

$- 12 \frac{- (19 x + 3)}{12} = - 12 \cdot - 5$

Factor $12$ out of $- 12$ .

$12 (- 1) \frac{- (19 x + 3)}{12} = - 12 \cdot - 5$

Cancel the common factor.

$12 \cdot - 1 \frac{- (19 x + 3)}{12} = - 12 \cdot - 5$

Rewrite the expression.

$- - (19 x + 3) = - 12 \cdot - 5$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (19 x + 3) = - 12 \cdot - 5$

Multiply $19 x + 3$ by $1$ .

$19 x + 3 = - 12 \cdot - 5$

Simplify the right side.

Multiply $- 12$ by $- 5$ .

$19 x + 3 = 60$

Step 5

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

Subtract $3$ from both sides of the equation.

$19 x = 60 - 3$

Subtract $3$ from $60$ .

$19 x = 57$

Step 6

Divide each term in $19 x = 57$ by $19$ and simplify.

Divide each term in $19 x = 57$ by $19$ .

$\frac{19 x}{19} = \frac{57}{19}$

Simplify the left side.

Cancel the common factor of $19$ .

Cancel the common factor.

$\frac{19 x}{19} = \frac{57}{19}$

Divide $x$ by $1$ .

$x = \frac{57}{19}$

Simplify the right side.

Divide $57$ by $19$ .

$x = 3$

Solve for x (x-5)/6+(3x+7)/12=2x-5

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