Let 2x-7y+4z-11=0 and -3x-5y+4z-3=0` are two planes .If planes ax+by+cz-7=0 passes through the line of intersection of given planes and point (-2,1,3) , then find the value of 2a+b+c+7

Answer

To solve the problem, we need to find the value of \(2a + b + c + 7\) given that the plane \(ax + by + cz - 7 = 0\) passes through the line of intersection of the two given planes and the point \((-2, 1, 3)\).

### Step 1: Write the equations of the given planes
The equations of the two planes are:
1. \(P_1: 2x - 7y + 4z - 11 = 0\)
2. \(P_2: -3x - 5y + 4z - 3 = 0\)

### Step 2: Find the equation of the line of intersection of the planes
The equation of the plane that passes through the line of intersection of \(P_1\) and \(P_2\) can be expressed as:
\[ P = P_1 + \lambda P_2 = 0 \]
Substituting the equations:
\[ (2x - 7y + 4z - 11) + \lambda(-3x - 5y + 4z - 3) = 0 \]

### Step 3: Combine the equations
Combining the equations gives:
\[ (2 - 3\lambda)x + (-7 - 5\lambda)y + (4 + 4\lambda)z - (11 + 3\lambda) = 0 \]
This can be rewritten as:
\[ (2 - 3\lambda)x + (-7 - 5\lambda)y + (4 + 4\lambda)z - (11 + 3\lambda) = 0 \]

### Step 4: Substitute the point \((-2, 1, 3)\)
Since the plane \(ax + by + cz - 7 = 0\) passes through the point \((-2, 1, 3)\), we substitute these coordinates into the equation:
\[ (2 - 3\lambda)(-2) + (-7 - 5\lambda)(1) + (4 + 4\lambda)(3) - (11 + 3\lambda) = 0 \]

### Step 5: Simplify the equation
Expanding this gives:
\[ -4 + 6\lambda - 7 - 5\lambda + 12 + 12\lambda - 11 - 3\lambda = 0 \]
Combining like terms:
\[ (6\lambda - 5\lambda + 12\lambda - 3\lambda) + (-4 - 7 - 11 + 12) = 0 \]
This simplifies to:
\[ 10\lambda - 10 = 0 \]

### Step 6: Solve for \(\lambda\)
From \(10\lambda - 10 = 0\), we find:
\[ \lambda = 1 \]

### Step 7: Substitute \(\lambda\) back into the equation of the plane
Substituting \(\lambda = 1\) back into the equation of the plane gives:
\[ (2 - 3(1))x + (-7 - 5(1))y + (4 + 4(1))z - (11 + 3(1)) = 0 \]
This simplifies to:
\[ (-1)x + (-12)y + (8)z - (14) = 0 \]
Thus, the equation of the plane is:
\[ -x - 12y + 8z - 14 = 0 \]

### Step 8: Identify coefficients \(a\), \(b\), and \(c\)
From the equation \(-x - 12y + 8z - 14 = 0\), we have:
- \(a = -1\)
- \(b = -12\)
- \(c = 8\)

### Step 9: Calculate \(2a + b + c + 7\)
Now, we calculate:
\[ 2a + b + c + 7 = 2(-1) + (-12) + 8 + 7 \]
Calculating this gives:
\[ -2 - 12 + 8 + 7 = -2 - 12 + 15 = 1 \]

### Final Answer
Thus, the value of \(2a + b + c + 7\) is:
\[ \boxed{1} \]