Find the distance of the point `(-1, 1)` from the line `12(x+6)=5(y-2)`.

Answer

To find the distance of the point \((-1, 1)\) from the line given by the equation \(12(x + 6) = 5(y - 2)\), we will follow these steps:

### Step 1: Simplify the line equation
Start by simplifying the equation of the line:

\[ 12(x + 6) = 5(y - 2) \]

Expanding both sides:

\[ 12x + 72 = 5y - 10 \]

Rearranging this into the standard form \(Ax + By + C = 0\):

\[ 12x - 5y + 82 = 0 \]

### Step 2: Identify coefficients
From the equation \(12x - 5y + 82 = 0\), we can identify the coefficients:
- \(A = 12\)
- \(B = -5\)
- \(C = 82\)

### Step 3: Use the distance formula
The distance \(d\) from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) is given by the formula:

\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

Substituting \((x_1, y_1) = (-1, 1)\):

\[ d = \frac{|12(-1) + (-5)(1) + 82|}{\sqrt{12^2 + (-5)^2}} \]

### Step 4: Calculate the numerator
Calculating the numerator:

\[ 12(-1) = -12 \]
\[ -5(1) = -5 \]
\[ -12 - 5 + 82 = 65 \]

So, the absolute value is:

\[ |65| = 65 \]

### Step 5: Calculate the denominator
Calculating the denominator:

\[ \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 6: Final calculation of distance
Now substitute back into the distance formula:

\[ d = \frac{65}{13} = 5 \]

Thus, the distance of the point \((-1, 1)\) from the line is:

\[ \boxed{5} \]

---