Algebra Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Multiply both sides by $3$ .

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

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

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Simplify the left side.

Tap for more steps...

Step 1.2.1.1

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

Tap for more steps...

Step 1.2.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.1.1.1.1

Cancel the common factor.

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

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

Step 1.2.1.1.1.2

Rewrite the expression.

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

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

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

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

Step 1.2.1.1.2

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

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

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

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

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

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

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

Step 1.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.2.1

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

Tap for more steps...

Step 1.2.2.1.1

Apply the distributive property.

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

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

Step 1.2.2.1.2

Simplify the expression.

Tap for more steps...

Step 1.2.2.1.2.1

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

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

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

Step 1.2.2.1.2.2

Multiply $3$ by $- 1$ .

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

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

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

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

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

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

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

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

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

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

Step 1.3

Solve for $y$ .

Tap for more steps...

Step 1.3.1

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

Tap for more steps...

Step 1.3.1.1

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

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

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

Step 1.3.1.2

Add $5 y$ and $3 y$ .

$- 3 x + 8 y = 3 x$

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

$- 3 x + 8 y = 3 x$

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

Step 1.3.2

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

Tap for more steps...

Step 1.3.2.1

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

$8 y = 3 x + 3 x$

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

Step 1.3.2.2

Add $3 x$ and $3 x$ .

$8 y = 6 x$

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

$8 y = 6 x$

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

Step 1.3.3

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

Tap for more steps...

Step 1.3.3.1

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

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

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

Step 1.3.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.3.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 1.3.3.2.1.1

Cancel the common factor.

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

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

Step 1.3.3.2.1.2

Divide $y$ by $1$ .

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

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

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

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

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

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

Step 1.3.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.3.1

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

Tap for more steps...

Step 1.3.3.3.1.1

Factor $2$ out of $6 x$ .

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

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

Step 1.3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.3.3.3.1.2.1

Factor $2$ out of $8$ .

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

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

Step 1.3.3.3.1.2.2

Cancel the common factor.

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

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

Step 1.3.3.3.1.2.3

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.

Tap for more steps...

Step 2.1

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

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Simplify the left side.

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Simplify the numerator.

Tap for more steps...

Step 2.2.1.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.1.1.1

Factor $2$ out of $- 2$ .

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

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

Step 2.2.1.1.1.1.2

Factor $2$ out of $4$ .

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

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

Step 2.2.1.1.1.1.3

Cancel the common factor.

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

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

Step 2.2.1.1.1.1.4

Rewrite the expression.

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

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

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

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

Step 2.2.1.1.1.2

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

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

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

Step 2.2.1.1.1.3

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

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

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

Step 2.2.1.1.1.4

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

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

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

Step 2.2.1.1.1.5

Combine the numerators over the common denominator.

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

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

Step 2.2.1.1.1.6

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

Tap for more steps...

Step 2.2.1.1.1.6.1

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

Tap for more steps...

Step 2.2.1.1.1.6.1.1

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

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

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

Step 2.2.1.1.1.6.1.2

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

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

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

Step 2.2.1.1.1.6.1.3

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

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

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

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

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

Step 2.2.1.1.1.6.2

Subtract $1$ from $2$ .

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

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

Step 2.2.1.1.1.6.3

Multiply $3$ by $1$ .

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

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

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

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

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

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

Step 2.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 2.2.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1.3.1

Factor $3$ out of $3 x$ .

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

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

Step 2.2.1.1.3.2

Factor $3$ out of $6$ .

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

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

Step 2.2.1.1.3.3

Cancel the common factor.

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

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

Step 2.2.1.1.3.4

Rewrite the expression.

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

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

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

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

Step 2.2.1.1.4

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

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

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

Step 2.2.1.1.5

Multiply $2$ by $2$ .

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

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

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

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

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

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

Step 2.2.2

Simplify the right side.

Tap for more steps...

Step 2.2.2.1

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

Tap for more steps...

Step 2.2.2.1.1

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

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

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

Step 2.2.2.1.2

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

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

Step 3.1.2

Combine the numerators over the common denominator.

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

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

Step 3.1.3

Subtract $9 x$ from $- x$ .

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

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

Step 3.1.4

Simplify each term.

Tap for more steps...

Step 3.1.4.1

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

Tap for more steps...

Step 3.1.4.1.1

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

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

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

Step 3.1.4.1.2

Cancel the common factors.

Tap for more steps...

Step 3.1.4.1.2.1

Factor $2$ out of $4$ .

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

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

Step 3.1.4.1.2.2

Cancel the common factor.

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

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

Step 3.1.4.1.2.3

Rewrite the expression.

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

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

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

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

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

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

Step 3.1.4.2

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

Subtract $12$ from both sides of the equation.

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

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

Step 3.2.2

Subtract $12$ from $2$ .

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

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

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

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

Step 3.3

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

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

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

Step 3.4

Simplify both sides of the equation.

Tap for more steps...

Step 3.4.1

Simplify the left side.

Tap for more steps...

Step 3.4.1.1

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

Tap for more steps...

Step 3.4.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.1.1.1.1

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

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

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

Step 3.4.1.1.1.2

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

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

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

Step 3.4.1.1.1.3

Factor $2$ out of $- 2$ .

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

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

Step 3.4.1.1.1.4

Cancel the common factor.

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

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

Step 3.4.1.1.1.5

Rewrite the expression.

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

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

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

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

Step 3.4.1.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.4.1.1.2.1

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

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

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

Step 3.4.1.1.2.2

Cancel the common factor.

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

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

Step 3.4.1.1.2.3

Rewrite the expression.

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

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

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

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

Step 3.4.1.1.3

Multiply.

Tap for more steps...

Step 3.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

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

Step 3.4.1.1.3.2

Multiply $x$ by $1$ .

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

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

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

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

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

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

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

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

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

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

Tap for more steps...

Step 3.4.2.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.4.2.1.1.1

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

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

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

Step 3.4.2.1.1.2

Factor $5$ out of $- 10$ .

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

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

Step 3.4.2.1.1.3

Cancel the common factor.

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

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

Step 3.4.2.1.1.4

Rewrite the expression.

$x = - 2 \cdot - 2$

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

$x = - 2 \cdot - 2$

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

Step 3.4.2.1.2

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.

Tap for more steps...

Step 4.1

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

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

$x = 4$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

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

$x = 4$

Step 4.2.1.2

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