Task B-2.7. Today is Valentine's Day, and Valentino wants to go out with his girlfriend Lorna. With his 120 kuna, he can buy several roses and two lemonades. The price of one rose is equal to the price of one lemonade. Since they have a math test tomorrow, they will postpone the outing until Saturday. On Saturday, the price of one rose will decrease by 7 kuna. Valentino wants to buy the same number of roses, and instead of lemonades, he wants to buy a jumbo pizza and a Coca-Cola, which together cost 78 kuna. However, he would be short of 6 kuna. How many roses will Valentino buy for Lorna? At what price will he buy the roses on Saturday?
Solution. Let $x$ be the number of roses, and $y$ be the price of one rose on Valentine's Day. From the conditions of the problem, we have

$$
\begin{aligned}
x y+2 y & =120 \\
x(y-7)+78 & =126
\end{aligned}
$$

We solve the system

$$
\left\{\begin{array}{l}
x y+2 y=120 \\
x y-7 x=48
\end{array}\right.
$$

If we express $y=\frac{120}{x+2}$ from the first equation and substitute it into the second, we get

$$
x \cdot \frac{120}{x+2}-7 x=48
$$

which simplifies to the quadratic equation

$$
7 x^{2}-58 x+96=0
$$

Its integer solution is $x=6$.

Then $y=\frac{120}{6+2}=15$.

Valentino will buy 6 roses, and if he buys them on Saturday, he will pay 8 kn per piece.