Solve k*50*5/x^2=kx*30*5/(80-x)^2

$$\frac { k \times 50 \times 5 } { x ^ { 2 } } = \frac { k x \cdot 30 \times 5 } { ( 80 - x ) ^ { 2 } }$$

Solve for k

$$\left\{\begin{matrix}k=0\text{, }&x\neq 0\text{ and }x\neq 80\\k\in \mathrm{R}\text{, }&x=\frac{5\sqrt[3]{1728\sqrt{323}+30673}+5\sqrt[3]{30673-1728\sqrt{323}}+5}{9}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}\left(x-80\right)^{2}, the least common multiple of x^{2},\left(80-x\right)^{2}.

$$\left(x-80\right)^{2}k\times 50\times 5=x^{2}kx\times 30\times 5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-80\right)^{2}.

$$\left(x^{2}-160x+6400\right)k\times 50\times 5=x^{2}kx\times 30\times 5$$

Multiply 50 and 5 to get 250.

$$\left(x^{2}-160x+6400\right)k\times 250=x^{2}kx\times 30\times 5$$

Use the distributive property to multiply x^{2}-160x+6400 by k.

$$\left(x^{2}k-160xk+6400k\right)\times 250=x^{2}kx\times 30\times 5$$

Use the distributive property to multiply x^{2}k-160xk+6400k by 250.

$$250x^{2}k-40000xk+1600000k=x^{2}kx\times 30\times 5$$

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

$$250x^{2}k-40000xk+1600000k=x^{3}k\times 30\times 5$$

Multiply 30 and 5 to get 150.

$$250x^{2}k-40000xk+1600000k=x^{3}k\times 150$$

Subtract x^{3}k\times 150 from both sides.

$$250x^{2}k-40000xk+1600000k-x^{3}k\times 150=0$$

Multiply -1 and 150 to get -150.

$$250x^{2}k-40000xk+1600000k-150x^{3}k=0$$

Combine all terms containing k.

$$\left(250x^{2}-40000x+1600000-150x^{3}\right)k=0$$

The equation is in standard form.

$$\left(1600000-40000x+250x^{2}-150x^{3}\right)k=0$$

Divide 0 by 250x^{2}-40000x+1600000-150x^{3}.

$$k=0$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{5\left(\sqrt[3]{1728\sqrt{323}+30673}+\sqrt[3]{30673-1728\sqrt{323}}+1\right)}{9}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\setminus 0,80\text{, }&k=0\end{matrix}\right.$$