Solve for x (3x)/2+(2x)/3=(1+3x)/2

$\frac{3 x}{2} + \frac{2 x}{3} = \frac{1 + 3 x}{2}$

Step 1

Multiply both sides by $2$ .

$(\frac{3 x}{2} + \frac{2 x}{3}) \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

Step 2

Simplify.

Simplify the left side.

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

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

$(\frac{3 x}{2} \cdot \frac{3}{3} + \frac{2 x}{3}) \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

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

$(\frac{3 x}{2} \cdot \frac{3}{3} + \frac{2 x}{3} \cdot \frac{2}{2}) \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

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

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

$(\frac{3 x \cdot 3}{2 \cdot 3} + \frac{2 x}{3} \cdot \frac{2}{2}) \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

Multiply $2$ by $3$ .

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

$\frac{3 x \cdot 3 + 2 x \cdot 2}{6} \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

Simplify the numerator.

Factor $x$ out of $3 x \cdot 3 + 2 x \cdot 2$ .

Factor $x$ out of $3 x \cdot 3$ .

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

Factor $x$ out of $2 x \cdot 2$ .

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

Factor $x$ out of $x (3 \cdot 3) + x (2 \cdot 2)$ .

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

Multiply $3$ by $3$ .

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

Multiply $2$ by $2$ .

$\frac{x (9 + 4)}{6} \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

Add $9$ and $4$ .

$\frac{x \cdot 13}{6} \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{x \cdot 13}{2 (3)} \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

Cancel the common factor.

$\frac{x \cdot 13}{2 \cdot 3} \cdot 2 = \frac{1 + 3 x}{2} \cdot 2$

Rewrite the expression.

$\frac{x \cdot 13}{3} = \frac{1 + 3 x}{2} \cdot 2$

Move $13$ to the left of $x$ .

$\frac{13 x}{3} = \frac{1 + 3 x}{2} \cdot 2$

Simplify the right side.

Simplify $\frac{1 + 3 x}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{13 x}{3} = \frac{1 + 3 x}{2} \cdot 2$

Rewrite the expression.

$\frac{13 x}{3} = 1 + 3 x$

Reorder $1$ and $3 x$ .

$\frac{13 x}{3} = 3 x + 1$

Step 3

Solve for $x$ .

Multiply both sides by $3$ .

$\frac{13 x}{3} \cdot 3 = (3 x + 1) \cdot 3$

Simplify.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{13 x}{3} \cdot 3 = (3 x + 1) \cdot 3$

Rewrite the expression.

$13 x = (3 x + 1) \cdot 3$

Simplify the right side.

Simplify $(3 x + 1) \cdot 3$ .

Apply the distributive property.

$13 x = 3 x \cdot 3 + 1 \cdot 3$

Multiply.

Multiply $3$ by $3$ .

$13 x = 9 x + 1 \cdot 3$

Multiply $3$ by $1$ .

$13 x = 9 x + 3$

Solve for $x$ .

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

Subtract $9 x$ from both sides of the equation.

$13 x - 9 x = 3$

Subtract $9 x$ from $13 x$ .

$4 x = 3$

Divide each term in $4 x = 3$ by $4$ and simplify.

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{4}$

Decimal Form:

$x = 0.75$

Solve for x (3x)/2+(2x)/3=(1+3x)/2

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