If `y=3[x]+1=4[x-1]-10,` then find the value of `[x+2y]`.

Explanation

To solve the equation \( y = 3[x] + 1 = 4[x - 1] - 10 \), we will follow these steps:

### Step 1: Set up the equation
We have two expressions for \( y \):
1. \( y = 3[x] + 1 \)
2. \( y = 4[x - 1] - 10 \)

We can set these two expressions equal to each other:
\[ 3[x] + 1 = 4[x - 1] - 10 \]

### Step 2: Simplify the equation
Now, let's simplify the equation:
\[ 3[x] + 1 = 4[x] - 4 - 10 \]
This simplifies to:
\[ 3[x] + 1 = 4[x] - 14 \]

### Step 3: Rearrange the equation
Next, we can rearrange the equation to isolate terms involving \( [x] \):
\[ 3[x] + 1 + 14 = 4[x] \]
\[ 3[x] + 15 = 4[x] \]

### Step 4: Solve for \( [x] \)
Now, we can isolate \( [x] \) by moving \( 3[x] \) to the right side:
\[ 15 = 4[x] - 3[x] \]
\[ 15 = [x] \]

### Step 5: Determine the range of \( x \)
Since \( [x] = 15 \), this means:
\[ 15 \leq x < 16 \]

### Step 6: Calculate \( y \)
Now, we can substitute \( [x] \) back into the equation for \( y \):
\[ y = 3[x] + 1 = 3(15) + 1 = 45 + 1 = 46 \]

### Step 7: Find \( [x + 2y] \)
Now, we need to calculate \( [x + 2y] \):
\[ x + 2y = x + 2(46) = x + 92 \]
Since \( 15 \leq x < 16 \), we can find the range of \( x + 92 \):
\[ 15 + 92 \leq x + 92 < 16 + 92 \]
\[ 107 \leq x + 92 < 108 \]

### Step 8: Determine \( [x + 2y] \)
Thus, the greatest integer function of \( x + 2y \) is:
\[ [x + 2y] = [x + 92] = 107 \]

### Final Answer
The value of \( [x + 2y] \) is \( 107 \).

---