Solve y=13k^2+3/9-9kx-9k^2+13/4k^2+9k

$$y = \frac { 13 k ^ { 2 } + 3 } { 9 - 9 k } x - \frac { 9 k ^ { 2 } + 13 } { 4 k ^ { 2 } + 9 k }$$

Solve for x

$$x=-\frac{9\left(k-1\right)\left(4yk^{2}+9ky+9k^{2}+13\right)}{k\left(4k+9\right)\left(13k^{2}+3\right)}$$

$$k\neq 1\text{ and }k\neq -\frac{9}{4}\text{ and }k\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 9k\left(k-1\right)\left(4k+9\right), the least common multiple of 9-9k,4k^{2}+9k.

$$y\times 9k\left(k-1\right)\left(4k+9\right)=\left(-9k-4k^{2}\right)\left(13k^{2}+3\right)x-\left(9k-9\right)\left(9k^{2}+13\right)$$

Use the distributive property to multiply y\times 9k by k-1.

$$\left(9yk^{2}-y\times 9k\right)\left(4k+9\right)=\left(-9k-4k^{2}\right)\left(13k^{2}+3\right)x-\left(9k-9\right)\left(9k^{2}+13\right)$$

Multiply -1 and 9 to get -9.

$$\left(9yk^{2}-9yk\right)\left(4k+9\right)=\left(-9k-4k^{2}\right)\left(13k^{2}+3\right)x-\left(9k-9\right)\left(9k^{2}+13\right)$$

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

$$36yk^{3}+45yk^{2}-81yk=\left(-9k-4k^{2}\right)\left(13k^{2}+3\right)x-\left(9k-9\right)\left(9k^{2}+13\right)$$

Use the distributive property to multiply -9k-4k^{2} by 13k^{2}+3.

$$36yk^{3}+45yk^{2}-81yk=\left(-117k^{3}-27k-52k^{4}-12k^{2}\right)x-\left(9k-9\right)\left(9k^{2}+13\right)$$

Use the distributive property to multiply -117k^{3}-27k-52k^{4}-12k^{2} by x.

$$36yk^{3}+45yk^{2}-81yk=-117k^{3}x-27kx-52k^{4}x-12k^{2}x-\left(9k-9\right)\left(9k^{2}+13\right)$$

Use the distributive property to multiply 9k-9 by 9k^{2}+13.

$$36yk^{3}+45yk^{2}-81yk=-117k^{3}x-27kx-52k^{4}x-12k^{2}x-\left(81k^{3}+117k-81k^{2}-117\right)$$

To find the opposite of 81k^{3}+117k-81k^{2}-117, find the opposite of each term.

$$36yk^{3}+45yk^{2}-81yk=-117k^{3}x-27kx-52k^{4}x-12k^{2}x-81k^{3}-117k+81k^{2}+117$$

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

$$-117k^{3}x-27kx-52k^{4}x-12k^{2}x-81k^{3}-117k+81k^{2}+117=36yk^{3}+45yk^{2}-81yk$$

Add 81k^{3} to both sides.

$$-117k^{3}x-27kx-52k^{4}x-12k^{2}x-117k+81k^{2}+117=36yk^{3}+45yk^{2}-81yk+81k^{3}$$

Add 117k to both sides.

$$-117k^{3}x-27kx-52k^{4}x-12k^{2}x+81k^{2}+117=36yk^{3}+45yk^{2}-81yk+81k^{3}+117k$$

Subtract 81k^{2} from both sides.

$$-117k^{3}x-27kx-52k^{4}x-12k^{2}x+117=36yk^{3}+45yk^{2}-81yk+81k^{3}+117k-81k^{2}$$

Subtract 117 from both sides.

$$-117k^{3}x-27kx-52k^{4}x-12k^{2}x=36yk^{3}+45yk^{2}-81yk+81k^{3}+117k-81k^{2}-117$$

Combine all terms containing x.

$$\left(-117k^{3}-27k-52k^{4}-12k^{2}\right)x=36yk^{3}+45yk^{2}-81yk+81k^{3}+117k-81k^{2}-117$$

The equation is in standard form.

$$\left(-52k^{4}-117k^{3}-12k^{2}-27k\right)x=36yk^{3}+45yk^{2}-81ky+81k^{3}-81k^{2}+117k-117$$

Divide both sides by -117k^{3}-27k-52k^{4}-12k^{2}.

$$\frac{\left(-52k^{4}-117k^{3}-12k^{2}-27k\right)x}{-52k^{4}-117k^{3}-12k^{2}-27k}=\frac{9\left(k-1\right)\left(4yk^{2}+9ky+9k^{2}+13\right)}{-52k^{4}-117k^{3}-12k^{2}-27k}$$

Dividing by -117k^{3}-27k-52k^{4}-12k^{2} undoes the multiplication by -117k^{3}-27k-52k^{4}-12k^{2}.

$$x=\frac{9\left(k-1\right)\left(4yk^{2}+9ky+9k^{2}+13\right)}{-52k^{4}-117k^{3}-12k^{2}-27k}$$

Divide 9\left(-1+k\right)\left(13+9yk+9k^{2}+4yk^{2}\right) by -117k^{3}-27k-52k^{4}-12k^{2}.

$$x=-\frac{9\left(k-1\right)\left(4yk^{2}+9ky+9k^{2}+13\right)}{k\left(4k+9\right)\left(13k^{2}+3\right)}$$