# What is the reason behind the demand function of a perfect complement good?

so I know that usually the income curve is equal to: $$x_1p_1 + x_2p_2 = m$$ if we rearrange this equation we get that the demand for good one ($$x_1$$) is equal to: $$x_1 = \frac{m-x_2p_2}{p_1}$$

None the less in my book it is said that:

The demand for good 1 is $$x_1 = \frac {m}{(p_1 + p_2)}$$

if anyone could explain me why $$x_1 = \frac {m}{(p_1 + p_2)}$$ instead of $$x_1 = \frac{m-x_2p_2}{p_1}$$ that would be greatly appreciated. many thanks in advance.

• What is an "income curve"? Budget lines and demand curves are not the same thing, please reread your book carefully. – Giskard Feb 11 at 17:28

Suppose you have $$M=20$$ to spend on shoes. Left shoes cost $$p_L=5$$ and right shoes cost $$p_R=5$$. A bundle, $$(x_L,x_R)$$ consists of $$x_L$$ pairs of left shoes and $$x_R$$ pairs of right shoes.
2. The price of a pair is $$p_P=p_L+p_R=10$$. Thus, you can afford to buy up to $$\frac{M}{p_P}=\frac{M}{p_L+p_R}$$ pairs (two pairs in our example).
3. Since you need one left shoe to make each pair, the number of left shoes you buy is simply the same as the number of pairs calculated in step 2: $$x_L=\frac{M}{p_L+p_R}$$, or two left shoes. Likewise, you need to buy $$x_R=\frac{M}{p_L+p_R}$$ right shoes.