Solve ky-3y=-kx+5x

$$k y - 3 y = - k x + 5 x$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=\frac{5x+3y}{x+y}\text{, }&y\neq -x\\k\in \mathrm{C}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \left(-k\right)x from both sides.

$$ky-3y-\left(-k\right)x=5x$$

Multiply -1 and -1 to get 1.

$$ky-3y+kx=5x$$

Add 3y to both sides.

$$ky+kx=5x+3y$$

Combine all terms containing k.

$$\left(y+x\right)k=5x+3y$$

The equation is in standard form.

$$\left(x+y\right)k=5x+3y$$

Divide both sides by x+y.

$$\frac{\left(x+y\right)k}{x+y}=\frac{5x+3y}{x+y}$$

Dividing by x+y undoes the multiplication by x+y.

$$k=\frac{5x+3y}{x+y}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{y\left(k-3\right)}{k-5}\text{, }&k\neq 5\\x\in \mathrm{C}\text{, }&y=0\text{ and }k=5\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(-k\right)x+5x=ky-3y$$

Reorder the terms.

$$-kx+5x=ky-3y$$

Combine all terms containing x.

$$\left(-k+5\right)x=ky-3y$$

The equation is in standard form.

$$\left(5-k\right)x=ky-3y$$

Divide both sides by -k+5.

$$\frac{\left(5-k\right)x}{5-k}=\frac{y\left(k-3\right)}{5-k}$$

Dividing by -k+5 undoes the multiplication by -k+5.

$$x=\frac{y\left(k-3\right)}{5-k}$$

Solve for k

$$\left\{\begin{matrix}k=\frac{5x+3y}{x+y}\text{, }&y\neq -x\\k\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \left(-k\right)x from both sides.

$$ky-3y-\left(-k\right)x=5x$$

Multiply -1 and -1 to get 1.

$$ky-3y+kx=5x$$

Add 3y to both sides.

$$ky+kx=5x+3y$$

Combine all terms containing k.

$$\left(y+x\right)k=5x+3y$$

The equation is in standard form.

$$\left(x+y\right)k=5x+3y$$

Divide both sides by x+y.

$$\frac{\left(x+y\right)k}{x+y}=\frac{5x+3y}{x+y}$$

Dividing by x+y undoes the multiplication by x+y.

$$k=\frac{5x+3y}{x+y}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y\left(k-3\right)}{k-5}\text{, }&k\neq 5\\x\in \mathrm{R}\text{, }&y=0\text{ and }k=5\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(-k\right)x+5x=ky-3y$$

Reorder the terms.

$$-kx+5x=ky-3y$$

Combine all terms containing x.

$$\left(-k+5\right)x=ky-3y$$

The equation is in standard form.

$$\left(5-k\right)x=ky-3y$$

Divide both sides by -k+5.

$$\frac{\left(5-k\right)x}{5-k}=\frac{y\left(k-3\right)}{5-k}$$

Dividing by -k+5 undoes the multiplication by -k+5.

$$x=\frac{y\left(k-3\right)}{5-k}$$