The solution of this system of equations is :

$$x=y=60$$

To solve the system of equations, we have:

1. $x + y = 120$

2. $2x + \frac{1}{2}y = 150$

To solve for $x$ and $y$, we can use substitution or elimination. Let's use the substitution method.

First, solve the first equation for $y$:

$$y = 120 - x$$

Now substitute this expression for $y$ into the second equation:

$$2x + \frac{1}{2}(120 - x) = 150$$

Simplify and solve for $x$:

$$2x + 60 - \frac{1}{2}x = 150$$

Combine like terms:

$$\frac{4x}{2} - \frac{x}{2} + 60 = 150$$

$$\frac{3x}{2} + 60 = 150$$

Subtract 60 from both sides:

$$\frac{3x}{2} = 90$$

Multiply both sides by 2 to solve for $x$:

$$3x = 180$$

Divide both sides by 3:

$$x = 60$$

Now substitute $x = 60$ back into the expression for $y$:

$$y = 120 - 60 = 60$$

Therefore, the solution to the system of equations is $x = 60$ and $y = 60$.

Let's check this result :

$$x + y = 60+60=120$$
$$2x + \frac{1}{2}y = 2\times60 + \frac{60}{2} = 120$$