Solve for x 2/3x+4/5=1/6x+3/10

$\frac{2}{3} x + \frac{4}{5} = \frac{1}{6} x + \frac{3}{10}$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$\frac{2 x}{3} + \frac{4}{5} = \frac{1}{6} x + \frac{3}{10}$

Step 2

Combine $\frac{1}{6}$ and $x$ .

$\frac{2 x}{3} + \frac{4}{5} = \frac{x}{6} + \frac{3}{10}$

Step 3

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

Subtract $\frac{x}{6}$ from both sides of the equation.

$\frac{2 x}{3} + \frac{4}{5} - \frac{x}{6} = \frac{3}{10}$

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

$\frac{2 x}{3} \cdot \frac{2}{2} - \frac{x}{6} + \frac{4}{5} = \frac{3}{10}$

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

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

$\frac{2 x \cdot 2}{3 \cdot 2} - \frac{x}{6} + \frac{4}{5} = \frac{3}{10}$

Multiply $3$ by $2$ .

$\frac{2 x \cdot 2}{6} - \frac{x}{6} + \frac{4}{5} = \frac{3}{10}$

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - x}{6} + \frac{4}{5} = \frac{3}{10}$

Simplify each term.

Simplify the numerator.

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

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

$\frac{x (2 \cdot 2) - x}{6} + \frac{4}{5} = \frac{3}{10}$

Factor $x$ out of $- x$ .

$\frac{x (2 \cdot 2) + x \cdot - 1}{6} + \frac{4}{5} = \frac{3}{10}$

Factor $x$ out of $x (2 \cdot 2) + x \cdot - 1$ .

$\frac{x (2 \cdot 2 - 1)}{6} + \frac{4}{5} = \frac{3}{10}$

Multiply $2$ by $2$ .

$\frac{x (4 - 1)}{6} + \frac{4}{5} = \frac{3}{10}$

Subtract $1$ from $4$ .

$\frac{x \cdot 3}{6} + \frac{4}{5} = \frac{3}{10}$

Cancel the common factor of $3$ and $6$ .

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

$\frac{3 \cdot x}{6} + \frac{4}{5} = \frac{3}{10}$

Cancel the common factors.

Factor $3$ out of $6$ .

$\frac{3 \cdot x}{3 \cdot 2} + \frac{4}{5} = \frac{3}{10}$

Cancel the common factor.

$\frac{3 \cdot x}{3 \cdot 2} + \frac{4}{5} = \frac{3}{10}$

Rewrite the expression.

$\frac{x}{2} + \frac{4}{5} = \frac{3}{10}$

Step 4

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

Subtract $\frac{4}{5}$ from both sides of the equation.

$\frac{x}{2} = \frac{3}{10} - \frac{4}{5}$

To write $- \frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{3}{10} - \frac{4}{5} \cdot \frac{2}{2}$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{5}$ by $\frac{2}{2}$ .

$\frac{x}{2} = \frac{3}{10} - \frac{4 \cdot 2}{5 \cdot 2}$

Multiply $5$ by $2$ .

$\frac{x}{2} = \frac{3}{10} - \frac{4 \cdot 2}{10}$

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{3 - 4 \cdot 2}{10}$

Simplify the numerator.

Multiply $- 4$ by $2$ .

$\frac{x}{2} = \frac{3 - 8}{10}$

Subtract $8$ from $3$ .

$\frac{x}{2} = \frac{- 5}{10}$

Cancel the common factor of $- 5$ and $10$ .

Factor $5$ out of $- 5$ .

$\frac{x}{2} = \frac{5 (- 1)}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$\frac{x}{2} = \frac{5 \cdot - 1}{5 \cdot 2}$

Cancel the common factor.

$\frac{x}{2} = \frac{5 \cdot - 1}{5 \cdot 2}$

Rewrite the expression.

$\frac{x}{2} = \frac{- 1}{2}$

Move the negative in front of the fraction.

$\frac{x}{2} = - \frac{1}{2}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - 1$

Solve for x 2/3x+4/5=1/6x+3/10

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