Suppose that `vec p,vecqand vecr` are three non- coplaner in `R^(3)` ,Let the components of a vector`vecs` along `vecp , vec q and vecr` be 4,3, and 5, respectively , if the components this vector `vec s` along `(-vecp+vec q +vecr),(vecp-vecq+vecr) and (-vecp-vecq+vecr)` are x, y and z , respectively , then the value of `2x+y+z` is

Explanation

To solve the problem, we need to find the value of \(2x + y + z\) given the components of the vector \(\vec{s}\) along the vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\).

### Step 1: Express the vector \(\vec{s}\)
Given that the components of the vector \(\vec{s}\) along \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) are 4, 3, and 5 respectively, we can express \(\vec{s}\) as:
\[ \vec{s} = 4\vec{p} + 3\vec{q} + 5\vec{r} \]

**Hint:** Remember that the components of a vector along other vectors can be expressed as a linear combination of those vectors.

### Step 2: Express \(\vec{s}\) in terms of new vectors
We need to find the components of \(\vec{s}\) along the new vectors:
1. \(-\vec{p} + \vec{q} + \vec{r}\)
2. \(\vec{p} - \vec{q} + \vec{r}\)
3. \(-\vec{p} - \vec{q} + \vec{r}\)

Let the components of \(\vec{s}\) along these vectors be \(x\), \(y\), and \(z\) respectively. We can write:
\[ \vec{s} = x(-\vec{p} + \vec{q} + \vec{r}) + y(\vec{p} - \vec{q} + \vec{r}) + z(-\vec{p} - \vec{q} + \vec{r}) \]

**Hint:** Set up the equation for \(\vec{s}\) using the new basis formed by the given vectors.

### Step 3: Expand the equation
Expanding the right-hand side gives:
\[ \vec{s} = x(-\vec{p}) + x\vec{q} + x\vec{r} + y\vec{p} - y\vec{q} + y\vec{r} + z(-\vec{p}) + z(-\vec{q}) + z\vec{r} \]
Combining like terms:
\[ \vec{s} = (-x + y - z)\vec{p} + (x - y - z)\vec{q} + (x + y + z)\vec{r} \]

**Hint:** Group the coefficients of \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) to compare with the original expression for \(\vec{s}\).

### Step 4: Set up equations by comparing coefficients
Now, we can equate the coefficients from both expressions:
1. \(-x + y - z = 4\) (coefficient of \(\vec{p}\))
2. \(x - y - z = 3\) (coefficient of \(\vec{q}\))
3. \(x + y + z = 5\) (coefficient of \(\vec{r}\))

**Hint:** Write down the system of equations clearly for easier manipulation.

### Step 5: Solve the system of equations
From the first equation:
\[ -x + y - z = 4 \quad \text{(1)} \]
From the second equation:
\[ x - y - z = 3 \quad \text{(2)} \]
From the third equation:
\[ x + y + z = 5 \quad \text{(3)} \]

Now we can solve these equations step by step.

### Step 6: Add equations (1) and (2)
Adding equations (1) and (2):
\[ (-x + y - z) + (x - y - z) = 4 + 3 \]
This simplifies to:
\[ -y - 2z = 7 \quad \text{(4)} \]

### Step 7: Add equations (2) and (3)
Adding equations (2) and (3):
\[ (x - y - z) + (x + y + z) = 3 + 5 \]
This simplifies to:
\[ 2x = 8 \implies x = 4 \]

**Hint:** Use substitution to simplify the equations further.

### Step 8: Substitute \(x\) back into equations
Substituting \(x = 4\) into equation (3):
\[ 4 + y + z = 5 \implies y + z = 1 \quad \text{(5)} \]

Substituting \(x = 4\) into equation (2):
\[ 4 - y - z = 3 \implies -y - z = -1 \implies y + z = 1 \quad \text{(consistent with (5))} \]

### Step 9: Solve for \(y\) and \(z\)
Now substitute \(y + z = 1\) into equation (4):
\[ -y - 2z = 7 \implies - (1 - z) - 2z = 7 \implies -1 + z - 2z = 7 \]
This simplifies to:
\[ -z = 8 \implies z = -8 \]
Now substitute \(z = -8\) back into \(y + z = 1\):
\[ y - 8 = 1 \implies y = 9 \]

### Step 10: Calculate \(2x + y + z\)
Now we have:
\[ x = 4, \quad y = 9, \quad z = -8 \]
Calculating \(2x + y + z\):
\[ 2(4) + 9 - 8 = 8 + 9 - 8 = 9 \]

### Final Answer
Thus, the value of \(2x + y + z\) is:
\[ \boxed{9} \]