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 start with the two expressions for \( y \):
\[ 3[x] + 1 = 2[x - 3] + 5 \]

### Step 2: Simplify the right side
First, simplify the right side of the equation:
\[ 2[x - 3] + 5 = 2[x] - 6 + 5 = 2[x] - 1 \]
Now the equation looks like this:
\[ 3[x] + 1 = 2[x] - 1 \]

### Step 3: Rearrange the equation
Next, we can rearrange the equation to isolate the greatest integer function:
\[ 3[x] - 2[x] = -1 - 1 \]
This simplifies to:
\[ [x] = -2 \]

### Step 4: Substitute back to find \( y \)
Now that we have \( [x] = -2 \), we can substitute this back into the expression for \( y \):
\[ y = 3[-2] + 1 = -6 + 1 = -5 \]

### Step 5: Calculate \( [x + y] \)
Now we need to find \( [x + y] \):
\[ x + y = [x] + y = -2 + (-5) = -7 \]

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