# How to manipulate these two equations to show equality?

I have this equation, and I know that the equality is true, but I don't understand why it is true. I am trying to manipulate it to turn the left side into the right side, but I don't know where to start. Does anyone have an idea?

$$x_1y_2 + x_2y_1 = (x_1+x_2)(y_1+y_2) - x_1y_1 - x_2y_2$$

Thanks

• Start with the right hand side and transform it into the left. – Daniel Fischer Sep 24 '13 at 20:39

The main key is the distributive law:

\begin{align*} (x_1+x_2)(y_1+y_2) &= x_1(y_1+y_2)+x_2(y_1+y_2) & \text{using right distributivity} \\ &= x_1y_1+x_1y_2+x_2y_1+x_2y_2 & \text{using left distributivity}. \end{align*}

This can be rearranged to obtain the desired identity.

It might be conceptually easier to see that the first step is valid by making a substitution $u=y_1+y_2$. \begin{align*} (x_1+x_2)(y_1+y_2) &= (x_1+x_2)u & \text{since } u=y_1+y_2 \\ &= x_1u+x_2u & \text{using right distributivity} \\ &= x_1(y_1+y_2)+x_2(y_1+y_2) & \text{since } u=y_1+y_2 \\ &= x_1y_1+x_1y_2+x_2y_1+x_2y_2 & \text{using left distributivity}. \end{align*}

if you re-arrange the term in parathesis it becomes:

$$(x_1+x_2)(y_1+y_2) = x_1y_1 + x_1y_2 + x_2y_1 + x_2y_2$$

try inserting that into your equation