## Task 4 - 070734

Given the equation

$$
\frac{x}{2}+\frac{x}{3}+7=x-\frac{3}{4}
$$

In this equation, the addend 7 is to be replaced by another number so that $x=11$ satisfies the equation.

What is this number?
(I) Suppose $a$ is a number as sought in the problem statement. Then $x=11$ satisfies the equation $\frac{x}{2}+\frac{x}{3}+a=x-\frac{3}{4}$, that is: The equation $\frac{11}{2}+\frac{11}{3}+a=11-\frac{3}{4}$ holds. From this, it follows that

$$
a=11-\frac{3}{4}-\frac{11}{2}-\frac{11}{3}=\frac{13}{12}
$$

Thus, the number $a=\frac{13}{12}$ at most has the required property.

(II) If $x=11$ and $a=\frac{13}{12}$, then

$$
\frac{x}{2}+\frac{x}{3}+a=\frac{11}{2}+\frac{11}{3}+\frac{13}{12}=\frac{41}{4} \Rightarrow x-\frac{3}{4}=\frac{41}{4}
$$

so that in the case $a=\frac{13}{12}$, the number $x=11$ satisfies equation (1).