Solve the System of Equations 12-(3x-2y)/6=3y+2 (5y-3x)/3=x-y

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

Step 1

Solve for $y$ in $\frac{5 y - 3 x}{3} = x - y$ .

Multiply both sides by $3$ .

$\frac{5 y - 3 x}{3} \cdot 3 = (x - y) \cdot 3$

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

Simplify.

Simplify the left side.

Simplify $\frac{5 y - 3 x}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{5 y - 3 x}{3} \cdot 3 = (x - y) \cdot 3$

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

Rewrite the expression.

$5 y - 3 x = (x - y) \cdot 3$

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

$5 y - 3 x = (x - y) \cdot 3$

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

Reorder $5 y$ and $- 3 x$ .

$- 3 x + 5 y = (x - y) \cdot 3$

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

$- 3 x + 5 y = (x - y) \cdot 3$

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

$- 3 x + 5 y = (x - y) \cdot 3$

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

Simplify the right side.

Simplify $(x - y) \cdot 3$ .

Apply the distributive property.

$- 3 x + 5 y = x \cdot 3 - y \cdot 3$

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

Simplify the expression.

Move $3$ to the left of $x$ .

$- 3 x + 5 y = 3 \cdot x - y \cdot 3$

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

Multiply $3$ by $- 1$ .

$- 3 x + 5 y = 3 x - 3 y$

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

$- 3 x + 5 y = 3 x - 3 y$

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

$- 3 x + 5 y = 3 x - 3 y$

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

$- 3 x + 5 y = 3 x - 3 y$

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

$- 3 x + 5 y = 3 x - 3 y$

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

Solve for $y$ .

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

Add $3 y$ to both sides of the equation.

$- 3 x + 5 y + 3 y = 3 x$

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

Add $5 y$ and $3 y$ .

$- 3 x + 8 y = 3 x$

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

$- 3 x + 8 y = 3 x$

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

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

Add $3 x$ to both sides of the equation.

$8 y = 3 x + 3 x$

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

Add $3 x$ and $3 x$ .

$8 y = 6 x$

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

$8 y = 6 x$

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

Divide each term in $8 y = 6 x$ by $8$ and simplify.

Divide each term in $8 y = 6 x$ by $8$ .

$\frac{8 y}{8} = \frac{6 x}{8}$

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

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{6 x}{8}$

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

Divide $y$ by $1$ .

$y = \frac{6 x}{8}$

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

$y = \frac{6 x}{8}$

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

$y = \frac{6 x}{8}$

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

Simplify the right side.

Cancel the common factor of $6$ and $8$ .

Factor $2$ out of $6 x$ .

$y = \frac{2 (3 x)}{8}$

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

Cancel the common factors.

Factor $2$ out of $8$ .

$y = \frac{2 (3 x)}{2 (4)}$

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

Cancel the common factor.

$y = \frac{2 (3 x)}{2 \cdot 4}$

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

Rewrite the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Step 2

Replace all occurrences of $y$ with $\frac{3 x}{4}$ in each equation.

Replace all occurrences of $y$ in $12 - \frac{3 x - 2 y}{6} = 3 y + 2$ with $\frac{3 x}{4}$ .

$12 - \frac{3 x - 2 \frac{3 x}{4}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Simplify $12 - \frac{3 x - 2 \frac{3 x}{4}}{6} = 3 (\frac{3 x}{4}) + 2$ .

Simplify the left side.

Simplify each term.

Simplify the numerator.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$12 - \frac{3 x + 2 (- 1) (\frac{3 x}{4})}{6} = 3 (\frac{3 x}{4}) + 2$

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

Factor $2$ out of $4$ .

$12 - \frac{3 x + 2 \cdot (- 1 \frac{3 x}{2 \cdot 2})}{6} = 3 (\frac{3 x}{4}) + 2$

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

Cancel the common factor.

$12 - \frac{3 x + 2 \cdot (- 1 \frac{3 x}{2 \cdot 2})}{6} = 3 (\frac{3 x}{4}) + 2$

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

Rewrite the expression.

$12 - \frac{3 x - 1 \frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

$12 - \frac{3 x - 1 \frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Rewrite $- 1 \frac{3 x}{2}$ as $- \frac{3 x}{2}$ .

$12 - \frac{3 x - \frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

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

$12 - \frac{3 x \cdot \frac{2}{2} - \frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

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

$12 - \frac{\frac{3 x \cdot 2}{2} - \frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Combine the numerators over the common denominator.

$12 - \frac{\frac{3 x \cdot 2 - 3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Rewrite $\frac{3 x \cdot 2 - 3 x}{2}$ in a factored form.

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

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

$12 - \frac{\frac{3 x (2) - 3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Factor $3 x$ out of $- 3 x$ .

$12 - \frac{\frac{3 x (2) + 3 x (- 1)}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Factor $3 x$ out of $3 x (2) + 3 x (- 1)$ .

$12 - \frac{\frac{3 x (2 - 1)}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

$12 - \frac{\frac{3 x (2 - 1)}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Subtract $1$ from $2$ .

$12 - \frac{\frac{3 x \cdot 1}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Multiply $3$ by $1$ .

$12 - \frac{\frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

$12 - \frac{\frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

$12 - \frac{\frac{3 x}{2}}{6} = 3 (\frac{3 x}{4}) + 2$

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

Multiply the numerator by the reciprocal of the denominator.

$12 - (\frac{3 x}{2} \cdot \frac{1}{6}) = 3 (\frac{3 x}{4}) + 2$

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$12 - (\frac{3 (x)}{2} \cdot \frac{1}{6}) = 3 (\frac{3 x}{4}) + 2$

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

Factor $3$ out of $6$ .

$12 - (\frac{3 (x)}{2} \cdot \frac{1}{3 (2)}) = 3 (\frac{3 x}{4}) + 2$

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

Cancel the common factor.

$12 - (\frac{3 x}{2} \cdot \frac{1}{3 \cdot 2}) = 3 (\frac{3 x}{4}) + 2$

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

Rewrite the expression.

$12 - (\frac{x}{2} \cdot \frac{1}{2}) = 3 (\frac{3 x}{4}) + 2$

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

$12 - (\frac{x}{2} \cdot \frac{1}{2}) = 3 (\frac{3 x}{4}) + 2$

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

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

$12 - \frac{x}{2 \cdot 2} = 3 (\frac{3 x}{4}) + 2$

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

Multiply $2$ by $2$ .

$12 - \frac{x}{4} = 3 (\frac{3 x}{4}) + 2$

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

$12 - \frac{x}{4} = 3 (\frac{3 x}{4}) + 2$

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

$12 - \frac{x}{4} = 3 (\frac{3 x}{4}) + 2$

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

Simplify the right side.

Multiply $3 (\frac{3 x}{4})$ .

Combine $3$ and $\frac{3 x}{4}$ .

$12 - \frac{x}{4} = \frac{3 (3 x)}{4} + 2$

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

Multiply $3$ by $3$ .

$12 - \frac{x}{4} = \frac{9 x}{4} + 2$

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

$12 - \frac{x}{4} = \frac{9 x}{4} + 2$

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

$12 - \frac{x}{4} = \frac{9 x}{4} + 2$

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

$12 - \frac{x}{4} = \frac{9 x}{4} + 2$

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

$12 - \frac{x}{4} = \frac{9 x}{4} + 2$

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

Step 3

Solve for $x$ in $12 - \frac{x}{4} = \frac{9 x}{4} + 2$ .

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

Subtract $\frac{9 x}{4}$ from both sides of the equation.

$12 - \frac{x}{4} - \frac{9 x}{4} = 2$

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

Combine the numerators over the common denominator.

$12 + \frac{- x - 9 x}{4} = 2$

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

Subtract $9 x$ from $- x$ .

$12 + \frac{- 10 x}{4} = 2$

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

Simplify each term.

Cancel the common factor of $- 10$ and $4$ .

Factor $2$ out of $- 10 x$ .

$12 + \frac{2 (- 5 x)}{4} = 2$

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

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

Cancel the common factor.

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

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

Rewrite the expression.

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

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

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

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

Subtract $12$ from both sides of the equation.

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

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

Subtract $12$ from $2$ .

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

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

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

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

Multiply both sides of the equation by $- \frac{2}{5}$ .

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

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{2}{5} (- \frac{5 x}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

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

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

Move the leading negative in $- \frac{5 x}{2}$ into the numerator.

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

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

Factor $2$ out of $- 2$ .

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

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

Cancel the common factor.

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

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

Rewrite the expression.

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

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

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

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

Cancel the common factor of $5$ .

Factor $5$ out of $- 5 x$ .

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

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

Cancel the common factor.

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

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

Rewrite the expression.

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

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

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

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

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

Multiply $x$ by $1$ .

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

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

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

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

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

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

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

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

Simplify the right side.

Simplify $- \frac{2}{5} \cdot - 10$ .

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{2}{5}$ into the numerator.

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

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

Factor $5$ out of $- 10$ .

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

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

Cancel the common factor.

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

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

Rewrite the expression.

$x = - 2 \cdot - 2$

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

$x = - 2 \cdot - 2$

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

Multiply $- 2$ by $- 2$ .

$x = 4$

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

$x = 4$

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

$x = 4$

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

$x = 4$

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

$x = 4$

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

Step 4

Replace all occurrences of $x$ with $4$ in each equation.

Replace all occurrences of $x$ in $y = \frac{3 x}{4}$ with $4$ .

$y = \frac{3 (4)}{4}$

$x = 4$

Simplify the right side.

Cancel the common factor of $4$ .

Cancel the common factor.

$y = \frac{3 \cdot 4}{4}$

$x = 4$

Divide $3$ by $1$ .

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

$y = 3$

$x = 4$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(4, 3)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, 3)$

Equation Form:

$x = 4, y = 3$

Step 7

Solve the System of Equations 12-(3x-2y)/6=3y+2 (5y-3x)/3=x-y

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