If `y=3[x]+1=2[x-3]+5,` find the value of [x+y]

Answer

To solve the equation \( y = 3[x] + 1 = 2[x - 3] + 5 \), we will follow these steps:

### Step 1: Set the equations equal to each other
We have two expressions for \( y \):
1. \( y = 3[x] + 1 \)
2. \( y = 2[x - 3] + 5 \)

Since both expressions are equal to \( y \), we can set them equal to each other:
\[ 3[x] + 1 = 2[x - 3] + 5 \]

### Step 2: Simplify the equation
Now, we simplify the right side:
\[ 2[x - 3] + 5 = 2[x] - 6 + 5 = 2[x] - 1 \]
So, our equation becomes:
\[ 3[x] + 1 = 2[x] - 1 \]

### Step 3: Rearrange the equation
Next, we will rearrange the equation to isolate \( [x] \):
\[ 3[x] - 2[x] = -1 - 1 \]
This simplifies to:
\[ [x] = -2 \]

### Step 4: Substitute back to find \( y \)
Now that we have \( [x] = -2 \), we can substitute this value back into one of the expressions for \( y \). Let's use the first expression:
\[ y = 3[-2] + 1 = -6 + 1 = -5 \]

### Step 5: Calculate \( [x + y] \)
Now we need to find \( [x + y] \):
\[ x + y = x + (-5) \]
Since \( [x] = -2 \), we can express \( x \) as:
\[ x = -2 + f \quad \text{(where \( 0 \leq f < 1 \))} \]
Thus,
\[ x + y = (-2 + f) - 5 = -7 + f \]
The greatest integer function of \( x + y \) is:
\[ [x + y] = [-7 + f] = -7 \quad \text{(since \( f < 1 \))} \]

### Final Answer
Thus, the value of \( [x + y] \) is:
\[ \boxed{-7} \]