Solve in Terms of the Arbitrary Variable y 5x+6y=3 , 10x+12y=6

$5 x + 6 y = 3$ , $10 x + 12 y = 6$

Simplify the right side.

Simplify $\frac{3}{5} - \frac{6 (\frac{1}{2} - \frac{5 x}{6})}{5}$ .

Combine the numerators over the common denominator.

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

$y = \frac{1}{2} - \frac{5 x}{6}$

Simplify each term.

Apply the distributive property.

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

$y = \frac{1}{2} - \frac{5 x}{6}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 6$ .

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

$y = \frac{1}{2} - \frac{5 x}{6}$

Cancel the common factor.

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

$y = \frac{1}{2} - \frac{5 x}{6}$

Rewrite the expression.

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

$y = \frac{1}{2} - \frac{5 x}{6}$

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

$y = \frac{1}{2} - \frac{5 x}{6}$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{5 x}{6}$ into the numerator.

$x = \frac{3 - 3 - 6 \frac{- 5 x}{6}}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

Factor $6$ out of $- 6$ .

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

$y = \frac{1}{2} - \frac{5 x}{6}$

Cancel the common factor.

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

$y = \frac{1}{2} - \frac{5 x}{6}$

Rewrite the expression.

$x = \frac{3 - 3 - 1 (- 5 x)}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = \frac{3 - 3 - 1 (- 5 x)}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

Multiply $- 5$ by $- 1$ .

$x = \frac{3 - 3 + 5 x}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = \frac{3 - 3 + 5 x}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

Reduce the expression by cancelling the common factors.

Subtract $3$ from $3$ .

$x = \frac{0 + 5 x}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

Add $0$ and $5 x$ .

$x = \frac{5 x}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

Cancel the common factor of $5$ .

Cancel the common factor.

$x = \frac{5 x}{5}$

$y = \frac{1}{2} - \frac{5 x}{6}$

Divide $x$ by $1$ .

$x = x$

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = x$

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = x$

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = x$

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = x$

$y = \frac{1}{2} - \frac{5 x}{6}$

Simplify the right side.

Multiply $5$ by $x$ .

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = x$

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = x$

The simplified system is the arbitrary solution of the original system of equations.

$x = x$

$y = \frac{1}{2} - \frac{5 x}{6}$

Reduce the system.

$y = \frac{1}{2} - \frac{5 x}{6}$

$x = x$

Simplify the right side.

Reorder $\frac{1}{2}$ and $- \frac{5 x}{6}$ .

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

$x = x$

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

$x = x$

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