Cutting cake into 5 equal pieces 
If a cake is cut into $5$ equal pieces, each piece would be $80$ grams
  heavier than when the cake is cut into $7$ equal pieces. How heavy is
  the cake?

How would I solve this problem? Do I have to try to find an algebraic expression for this? $5x = 7y + 400$?

The first step is to turn the word problem into an equation; one-fifth of the cake is $80$ grams heavier than one-seventh of the cake, so one-fifth of the cake equals one-seventh of the cake plus 80. "The cake" (specifically its mass) is $x$, and we can work from there:
$$\dfrac{x}{5} = \dfrac{x}{7}+80$$
$$\dfrac{x}{5} - \dfrac{x}{7} = 80$$
Here comes the clever bit; multiply each fraction by a form of one that will give both fractions the same denominator:
$$\dfrac{7}{7}\cdot \dfrac{x}{5} - \dfrac{5}{5}\cdot\dfrac{x}{7} = 80$$
$$\dfrac{7x}{35} - \dfrac{5x}{35} = 80$$
$$\dfrac{2x}{35} = 80$$
$$2x = 2800$$
$$x = 1400$$
You can check your answer by plugging it into the original equation; if the two sides are indeed equal the answer is correct:
$$\dfrac{1400}{5} = \dfrac{1400}{7} + 80$$
$$280 = 200 + 80$$
$$\ \ \ \ \ \ \ \ \ \ \ \ 280 = 280 \ \ \text{<-- yay!}$$
