Solve for x (3x+1)/2+(5x-3)/3=(27+6x)/5

$\frac{3 x + 1}{2} + \frac{5 x - 3}{3} = \frac{27 + 6 x}{5}$

Step 1

Multiply both sides by $5$ .

$(\frac{3 x + 1}{2} + \frac{5 x - 3}{3}) \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Step 2

Simplify.

Simplify the left side.

Simplify $(\frac{3 x + 1}{2} + \frac{5 x - 3}{3}) \cdot 5$ .

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

$(\frac{3 x + 1}{2} \cdot \frac{3}{3} + \frac{5 x - 3}{3}) \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

To write $\frac{5 x - 3}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(\frac{3 x + 1}{2} \cdot \frac{3}{3} + \frac{5 x - 3}{3} \cdot \frac{2}{2}) \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

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

Multiply $\frac{3 x + 1}{2}$ by $\frac{3}{3}$ .

$(\frac{(3 x + 1) \cdot 3}{2 \cdot 3} + \frac{5 x - 3}{3} \cdot \frac{2}{2}) \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Multiply $2$ by $3$ .

$(\frac{(3 x + 1) \cdot 3}{6} + \frac{5 x - 3}{3} \cdot \frac{2}{2}) \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Multiply $\frac{5 x - 3}{3}$ by $\frac{2}{2}$ .

$(\frac{(3 x + 1) \cdot 3}{6} + \frac{(5 x - 3) \cdot 2}{3 \cdot 2}) \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Multiply $3$ by $2$ .

$(\frac{(3 x + 1) \cdot 3}{6} + \frac{(5 x - 3) \cdot 2}{6}) \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Combine the numerators over the common denominator.

$\frac{(3 x + 1) \cdot 3 + (5 x - 3) \cdot 2}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x \cdot 3 + 1 \cdot 3 + (5 x - 3) \cdot 2}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Multiply $3$ by $3$ .

$\frac{9 x + 1 \cdot 3 + (5 x - 3) \cdot 2}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Multiply $3$ by $1$ .

$\frac{9 x + 3 + (5 x - 3) \cdot 2}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Apply the distributive property.

$\frac{9 x + 3 + 5 x \cdot 2 - 3 \cdot 2}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Multiply $2$ by $5$ .

$\frac{9 x + 3 + 10 x - 3 \cdot 2}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Multiply $- 3$ by $2$ .

$\frac{9 x + 3 + 10 x - 6}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Add $9 x$ and $10 x$ .

$\frac{19 x + 3 - 6}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Subtract $6$ from $3$ .

$\frac{19 x - 3}{6} \cdot 5 = \frac{27 + 6 x}{5} \cdot 5$

Combine fractions.

Combine $\frac{19 x - 3}{6}$ and $5$ .

$\frac{(19 x - 3) \cdot 5}{6} = \frac{27 + 6 x}{5} \cdot 5$

Move $5$ to the left of $19 x - 3$ .

$\frac{5 (19 x - 3)}{6} = \frac{27 + 6 x}{5} \cdot 5$

Simplify the right side.

Simplify $\frac{27 + 6 x}{5} \cdot 5$ .

Factor $3$ out of $27 + 6 x$ .

Factor $3$ out of $27$ .

$\frac{5 (19 x - 3)}{6} = \frac{3 (9) + 6 x}{5} \cdot 5$

Factor $3$ out of $6 x$ .

$\frac{5 (19 x - 3)}{6} = \frac{3 (9) + 3 (2 x)}{5} \cdot 5$

Factor $3$ out of $3 (9) + 3 (2 x)$ .

$\frac{5 (19 x - 3)}{6} = \frac{3 (9 + 2 x)}{5} \cdot 5$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 (19 x - 3)}{6} = \frac{3 (9 + 2 x)}{5} \cdot 5$

Rewrite the expression.

$\frac{5 (19 x - 3)}{6} = 3 (9 + 2 x)$

Apply the distributive property.

$\frac{5 (19 x - 3)}{6} = 3 \cdot 9 + 3 (2 x)$

Simplify the expression.

Multiply $3$ by $9$ .

$\frac{5 (19 x - 3)}{6} = 27 + 3 (2 x)$

Multiply $2$ by $3$ .

$\frac{5 (19 x - 3)}{6} = 27 + 6 x$

Reorder $27$ and $6 x$ .

$\frac{5 (19 x - 3)}{6} = 6 x + 27$

Step 3

Solve for $x$ .

Multiply both sides by $6$ .

$\frac{5 (19 x - 3)}{6} \cdot 6 = (6 x + 27) \cdot 6$

Simplify.

Simplify the left side.

Simplify $\frac{5 (19 x - 3)}{6} \cdot 6$ .

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{5 (19 x - 3)}{6} \cdot 6 = (6 x + 27) \cdot 6$

Rewrite the expression.

$5 (19 x - 3) = (6 x + 27) \cdot 6$

Apply the distributive property.

$5 (19 x) + 5 \cdot - 3 = (6 x + 27) \cdot 6$

Multiply.

Multiply $19$ by $5$ .

$95 x + 5 \cdot - 3 = (6 x + 27) \cdot 6$

Multiply $5$ by $- 3$ .

$95 x - 15 = (6 x + 27) \cdot 6$

Simplify the right side.

Simplify $(6 x + 27) \cdot 6$ .

Apply the distributive property.

$95 x - 15 = 6 x \cdot 6 + 27 \cdot 6$

Multiply.

Multiply $6$ by $6$ .

$95 x - 15 = 36 x + 27 \cdot 6$

Multiply $27$ by $6$ .

$95 x - 15 = 36 x + 162$

Solve for $x$ .

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

Subtract $36 x$ from both sides of the equation.

$95 x - 15 - 36 x = 162$

Subtract $36 x$ from $95 x$ .

$59 x - 15 = 162$

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

Add $15$ to both sides of the equation.

$59 x = 162 + 15$

Add $162$ and $15$ .

$59 x = 177$

Divide each term in $59 x = 177$ by $59$ and simplify.

Divide each term in $59 x = 177$ by $59$ .

$\frac{59 x}{59} = \frac{177}{59}$

Simplify the left side.

Cancel the common factor of $59$ .

Cancel the common factor.

$\frac{59 x}{59} = \frac{177}{59}$

Divide $x$ by $1$ .

$x = \frac{177}{59}$

Simplify the right side.

Divide $177$ by $59$ .

$x = 3$

Solve for x (3x+1)/2+(5x-3)/3=(27+6x)/5

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