Solve y=2a^2+4a-6/a-1+2a^2+4a-6/3a+9quadb

$$y = \frac { 2 a ^ { 2 } + 4 a - 6 } { a - 1 } + \frac { 2 a ^ { 2 } + 4 a - 6 } { 3 a + 9 } \quad b$$

Solve for b

$$b=-\frac{3\left(6+2a-y\right)}{2\left(a-1\right)}$$

$$a\neq 1\text{ and }a\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 3\left(a-1\right)\left(a+3\right), the least common multiple of a-1,3a+9.

$$y\times 3\left(a-1\right)\left(a+3\right)=\left(3a+9\right)\left(2a^{2}+4a-6\right)+\left(a-1\right)\left(2a^{2}+4a-6\right)b$$

Use the distributive property to multiply y\times 3 by a-1.

$$\left(3ya-y\times 3\right)\left(a+3\right)=\left(3a+9\right)\left(2a^{2}+4a-6\right)+\left(a-1\right)\left(2a^{2}+4a-6\right)b$$

Multiply -1 and 3 to get -3.

$$\left(3ya-3y\right)\left(a+3\right)=\left(3a+9\right)\left(2a^{2}+4a-6\right)+\left(a-1\right)\left(2a^{2}+4a-6\right)b$$

Use the distributive property to multiply 3ya-3y by a+3 and combine like terms.

$$3ya^{2}+6ya-9y=\left(3a+9\right)\left(2a^{2}+4a-6\right)+\left(a-1\right)\left(2a^{2}+4a-6\right)b$$

Use the distributive property to multiply 3a+9 by 2a^{2}+4a-6 and combine like terms.

$$3ya^{2}+6ya-9y=6a^{3}+30a^{2}+18a-54+\left(a-1\right)\left(2a^{2}+4a-6\right)b$$

Use the distributive property to multiply a-1 by 2a^{2}+4a-6 and combine like terms.

$$3ya^{2}+6ya-9y=6a^{3}+30a^{2}+18a-54+\left(2a^{3}+2a^{2}-10a+6\right)b$$

Use the distributive property to multiply 2a^{3}+2a^{2}-10a+6 by b.

$$3ya^{2}+6ya-9y=6a^{3}+30a^{2}+18a-54+2a^{3}b+2a^{2}b-10ab+6b$$

Swap sides so that all variable terms are on the left hand side.

$$6a^{3}+30a^{2}+18a-54+2a^{3}b+2a^{2}b-10ab+6b=3ya^{2}+6ya-9y$$

Subtract 6a^{3} from both sides.

$$30a^{2}+18a-54+2a^{3}b+2a^{2}b-10ab+6b=3ya^{2}+6ya-9y-6a^{3}$$

Subtract 30a^{2} from both sides.

$$18a-54+2a^{3}b+2a^{2}b-10ab+6b=3ya^{2}+6ya-9y-6a^{3}-30a^{2}$$

Subtract 18a from both sides.

$$-54+2a^{3}b+2a^{2}b-10ab+6b=3ya^{2}+6ya-9y-6a^{3}-30a^{2}-18a$$

Add 54 to both sides.

$$2a^{3}b+2a^{2}b-10ab+6b=3ya^{2}+6ya-9y-6a^{3}-30a^{2}-18a+54$$

Combine all terms containing b.

$$\left(2a^{3}+2a^{2}-10a+6\right)b=3ya^{2}+6ya-9y-6a^{3}-30a^{2}-18a+54$$

The equation is in standard form.

$$\left(2a^{3}+2a^{2}-10a+6\right)b=3ya^{2}+6ay-9y-6a^{3}-30a^{2}-18a+54$$

Divide both sides by 2a^{3}+2a^{2}-10a+6.

$$\frac{\left(2a^{3}+2a^{2}-10a+6\right)b}{2a^{3}+2a^{2}-10a+6}=\frac{3\left(a-1\right)\left(a+3\right)\left(y-2a-6\right)}{2a^{3}+2a^{2}-10a+6}$$

Dividing by 2a^{3}+2a^{2}-10a+6 undoes the multiplication by 2a^{3}+2a^{2}-10a+6.

$$b=\frac{3\left(a-1\right)\left(a+3\right)\left(y-2a-6\right)}{2a^{3}+2a^{2}-10a+6}$$

Divide 3\left(-6+y-2a\right)\left(-1+a\right)\left(3+a\right) by 2a^{3}+2a^{2}-10a+6.

$$b=\frac{3\left(y-2a-6\right)}{2\left(a-1\right)}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{3y+2b-18}{2\left(b+3\right)}\text{, }&y\neq 8\text{ and }b\neq -3\text{ and }y\neq -\frac{8b}{3}\\a\in \mathrm{C}\setminus 1,-3\text{, }&y=8\text{ and }b=-3\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\frac{3y+2b-18}{2\left(b+3\right)}\text{, }&y\neq 8\text{ and }b\neq -3\text{ and }y\neq -\frac{8b}{3}\\a\in \mathrm{R}\setminus 1,-3\text{, }&y=8\text{ and }b=-3\end{matrix}\right.$$