Solve n:8x/5+9=13/7+2x/3

$$n : \frac { 8 x } { 5 } + 9 = \frac { 13 } { 7 } + \frac { 2 x } { 3 }$$

Solve for n

$$n=\frac{16x\left(7x-75\right)}{105}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 21, the least common multiple of 7,3.

$$\frac{105}{8}x^{-1}n+189=39+7\times 2x$$

Multiply 7 and 2 to get 14.

$$\frac{105}{8}x^{-1}n+189=39+14x$$

Subtract 189 from both sides.

$$\frac{105}{8}x^{-1}n=39+14x-189$$

Subtract 189 from 39 to get -150.

$$\frac{105}{8}x^{-1}n=-150+14x$$

Reorder the terms.

$$\frac{105}{8}\times \frac{1}{x}n=14x-150$$

Multiply both sides of the equation by 8x, the least common multiple of 8,x.

$$\frac{105}{8}\times 8\times 1n=14x\times 8x+8x\left(-150\right)$$

Multiply \frac{105}{8} and 8 to get 105.

$$105\times 1n=14x\times 8x+8x\left(-150\right)$$

Multiply 105 and 1 to get 105.

$$105n=14x\times 8x+8x\left(-150\right)$$

Multiply x and x to get x^{2}.

$$105n=14x^{2}\times 8+8x\left(-150\right)$$

Multiply 14 and 8 to get 112.

$$105n=112x^{2}+8x\left(-150\right)$$

Multiply 8 and -150 to get -1200.

$$105n=112x^{2}-1200x$$

Divide both sides by 105.

$$\frac{105n}{105}=\frac{16x\left(7x-75\right)}{105}$$

Dividing by 105 undoes the multiplication by 105.

$$n=\frac{16x\left(7x-75\right)}{105}$$

Divide 16x\left(-75+7x\right) by 105.

$$n=\frac{16x^{2}}{15}-\frac{80x}{7}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{\sqrt{735n+22500}}{28}+\frac{75}{14}\text{, }&\text{unconditionally}\\x=-\frac{\sqrt{735n+22500}}{28}+\frac{75}{14}\text{, }&n\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{735n+22500}}{28}+\frac{75}{14}\text{, }&n\geq -\frac{1500}{49}\\x=-\frac{\sqrt{735n+22500}}{28}+\frac{75}{14}\text{, }&n\neq 0\text{ and }n\geq -\frac{1500}{49}\end{matrix}\right.$$