3. Writing more advanced equations

The equations we saw in the previous chapter had only one unknown in them. Often, it is more natural to write equations in which there are two unknowns. We will study how this can be done in this chapter. Sometimes, a problem requires us to write equations in one unknown only, but the equations are somewhat more complicated than what you have seen so far. We will see some cases of this kind as well.

In this chapter, we will restrict ourselves to writing equations only. How to solve the equations will be discussed in subsequent chapters.

Let us begin with the second problem discussed in the Introduction.

Suppose 1 gm of rice contains .05 gm protein and .2 gm carbohydrate. Suppose 1 gm of wheat contains .08 gm protein and .16 gm carbohydrate. How much rice and wheat should I eat so that I get 45 gm protein and 180 gm carbohydrate?

There are two quantities unknown to us in this problem: the amount of rice and the amount of wheat that I should eat. So it is natural to use two symbols. So we might say:

Let $x$ denote the amount of rice (in grams) that I eat. Let $y$ denote the amount of wheat (in grams) that I eat.

Now all we need to do is to express the information given using $x$ and $y$. Since one gram of rice contains 0.05 gm of protein, $x$ grams of rice contain $x\times 0.05$ grams of protein. One gram of wheat contains 0.08 grams of protein. So $y$ grams of wheat contains $y\times 0.08$ grams of protein. But the total amount of protein I eat is 45 gram. So we must have:

$$0.05 x+0.08 y = 45$$

Similarly, one gram of rice contains 0.2 grams of carbohydrate. So $x$ grams must contain $0.2x$ grams. Likewise one gram of wheat contains 0.16 gram of carbohydrate. So $y$ grams contain $0.16y$ grams. Since the total carbohydrate consumption is 180 gram, it must be the case that:

$$0.2x+0.16y=180$$

The interpretation of the equations that we have written is still as before. The first equation effectively says, ``If I knew the what amount of rice to eat and what amount of wheat to eat and if these values were written in place of $x$ and $y$ in each of the algebraic equations, then the resulting arithmetic equations will be correct.''

How to solve the equations