Solve y^2+7y-18/C*14y^2/7y^2-14y=1

$$\frac { y ^ { 2 } + 7 y - 18 } { C } \cdot \frac { 14 y ^ { 2 } } { 7 y ^ { 2 } - 14 y } = 1$$

Solve for C

$$C=2y\left(y+9\right)$$

$$y\neq -9\text{ and }y\neq 0\text{ and }y\neq 2$$

Steps for Solving Linear Equation

Variable C cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 7Cy\left(y-2\right), the least common multiple of C,7y^{2}-14y.

$$7y\left(y-2\right)\left(y^{2}+7y-18\right)\times \frac{14y^{2}}{7y^{2}-14y}=7Cy\left(y-2\right)$$

Use the distributive property to multiply 7y by y-2.

$$\left(7y^{2}-14y\right)\left(y^{2}+7y-18\right)\times \frac{14y^{2}}{7y^{2}-14y}=7Cy\left(y-2\right)$$

Factor the expressions that are not already factored in \frac{14y^{2}}{7y^{2}-14y}.

$$\left(7y^{2}-14y\right)\left(y^{2}+7y-18\right)\times \frac{14y^{2}}{7y\left(y-2\right)}=7Cy\left(y-2\right)$$

Cancel out 7y in both numerator and denominator.

$$\left(7y^{2}-14y\right)\left(y^{2}+7y-18\right)\times \frac{2y}{y-2}=7Cy\left(y-2\right)$$

Express \left(7y^{2}-14y\right)\times \frac{2y}{y-2} as a single fraction.

$$\frac{\left(7y^{2}-14y\right)\times 2y}{y-2}\left(y^{2}+7y-18\right)=7Cy\left(y-2\right)$$

Use the distributive property to multiply \frac{\left(7y^{2}-14y\right)\times 2y}{y-2} by y^{2}+7y-18.

$$\frac{\left(7y^{2}-14y\right)\times 2y}{y-2}y^{2}+7\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Factor the expressions that are not already factored in \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}.

$$\frac{2\times 7\left(y-2\right)y^{2}}{y-2}y^{2}+7\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Cancel out y-2 in both numerator and denominator.

$$2\times 7y^{2}y^{2}+7\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Expand the expression.

$$14y^{2}y^{2}+7\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$14y^{4}+7\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Factor the expressions that are not already factored in \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}.

$$14y^{4}+7\times \frac{2\times 7\left(y-2\right)y^{2}}{y-2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Cancel out y-2 in both numerator and denominator.

$$14y^{4}+7\times 2\times 7y^{2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Expand the expression.

$$14y^{4}+7\times 14y^{2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Multiply 7 and 14 to get 98.

$$14y^{4}+98y^{2}y-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$14y^{4}+98y^{3}-18\times \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}=7Cy\left(y-2\right)$$

Factor the expressions that are not already factored in \frac{\left(7y^{2}-14y\right)\times 2y}{y-2}.

$$14y^{4}+98y^{3}-18\times \frac{2\times 7\left(y-2\right)y^{2}}{y-2}=7Cy\left(y-2\right)$$

Cancel out y-2 in both numerator and denominator.

$$14y^{4}+98y^{3}-18\times 2\times 7y^{2}=7Cy\left(y-2\right)$$

Expand the expression.

$$14y^{4}+98y^{3}-18\times 14y^{2}=7Cy\left(y-2\right)$$

Multiply -18 and 14 to get -252.

$$14y^{4}+98y^{3}-252y^{2}=7Cy\left(y-2\right)$$

Use the distributive property to multiply 7Cy by y-2.

$$14y^{4}+98y^{3}-252y^{2}=7Cy^{2}-14yC$$

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

$$7Cy^{2}-14yC=14y^{4}+98y^{3}-252y^{2}$$

Combine all terms containing C.

$$\left(7y^{2}-14y\right)C=14y^{4}+98y^{3}-252y^{2}$$

Divide both sides by 7y^{2}-14y.

$$\frac{\left(7y^{2}-14y\right)C}{7y^{2}-14y}=\frac{14\left(y-2\right)\left(y+9\right)y^{2}}{7y^{2}-14y}$$

Dividing by 7y^{2}-14y undoes the multiplication by 7y^{2}-14y.

$$C=\frac{14\left(y-2\right)\left(y+9\right)y^{2}}{7y^{2}-14y}$$

Divide 14\left(-2+y\right)\left(9+y\right)y^{2} by 7y^{2}-14y.

$$C=2y\left(y+9\right)$$

Variable C cannot be equal to 0.

$$C=2y\left(y+9\right)\text{, }C\neq 0$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{-\sqrt{2C+81}-9}{2}\text{, }&C\neq 0\\y=\frac{\sqrt{2C+81}-9}{2}\text{, }&C\neq 44\text{ and }C\neq 0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y=\frac{-\sqrt{2C+81}-9}{2}\text{, }&C\geq -\frac{81}{2}\text{ and }C\neq 0\\y=\frac{\sqrt{2C+81}-9}{2}\text{, }&C\neq 44\text{ and }C\geq -\frac{81}{2}\text{ and }C\neq 0\end{matrix}\right.$$