Solve AB+CD=AD+BC

$$A B + C D = A D + B C$$

Solve for A

$$\left\{\begin{matrix}\\A=C\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&B=D\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract AD from both sides.

$$AB+CD-AD=BC$$

Subtract CD from both sides.

$$AB-AD=BC-CD$$

Combine all terms containing A.

$$\left(B-D\right)A=BC-CD$$

Divide both sides by B-D.

$$\frac{\left(B-D\right)A}{B-D}=\frac{C\left(B-D\right)}{B-D}$$

Dividing by B-D undoes the multiplication by B-D.

$$A=\frac{C\left(B-D\right)}{B-D}$$

Divide C\left(B-D\right) by B-D.

$$A=C$$

Solve for B

$$\left\{\begin{matrix}\\B=D\text{, }&\text{unconditionally}\\B\in \mathrm{R}\text{, }&A=C\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract BC from both sides.

$$AB+CD-BC=AD$$

Subtract CD from both sides.

$$AB-BC=AD-CD$$

Combine all terms containing B.

$$\left(A-C\right)B=AD-CD$$

Divide both sides by A-C.

$$\frac{\left(A-C\right)B}{A-C}=\frac{D\left(A-C\right)}{A-C}$$

Dividing by A-C undoes the multiplication by A-C.

$$B=\frac{D\left(A-C\right)}{A-C}$$

Divide D\left(A-C\right) by A-C.

$$B=D$$