What is the solution of the following simultaneous equations?x+y+z=6,x+2y+3z=14,x+3y+z=10

Options

A. x=1,y=2,z=4
B. x=1,y=2,z=3
C. x=2,y=1,z=3
D. none of these

Answer

To solve the simultaneous equations:

1. **Equations Given:**
\[ \begin{align*} (1) & \quad x + y + z = 6 \\ (2) & \quad x + 2y + 3z = 14 \\ (3) & \quad x + 3y + z = 10 \end{align*} \]

2. **Step 1: Add the three equations together.**
\[ (x + y + z) + (x + 2y + 3z) + (x + 3y + z) = 6 + 14 + 10 \]
Simplifying the left side:
\[ 3x + (y + 2y + 3y) + (z + 3z + z) = 30 \]
This gives:
\[ 3x + 6y + 5z = 30 \quad \text{(4)} \]

3. **Step 2: Rearranging equation (4) for \(x + 2y\).**
\[ 3x + 6y = 30 - 5z \]
Dividing everything by 3:
\[ x + 2y = 10 - \frac{5z}{3} \quad \text{(5)} \]

4. **Step 3: Substitute equation (5) into equation (2).**
From equation (2):
\[ x + 2y + 3z = 14 \]
Substitute \(x + 2y\) from (5):
\[ (10 - \frac{5z}{3}) + 3z = 14 \]
Simplifying this:
\[ 10 - \frac{5z}{3} + 3z = 14 \]
Rearranging gives:
\[ 3z - \frac{5z}{3} = 14 - 10 \]
\[ 3z - \frac{5z}{3} = 4 \]
To combine the terms, convert \(3z\) to a fraction:
\[ \frac{9z}{3} - \frac{5z}{3} = 4 \]
This simplifies to:
\[ \frac{4z}{3} = 4 \]
Multiplying both sides by 3:
\[ 4z = 12 \quad \Rightarrow \quad z = 3 \]

5. **Step 4: Substitute \(z = 3\) back into equation (1).**
From equation (1):
\[ x + y + z = 6 \]
Substitute \(z\):
\[ x + y + 3 = 6 \]
This simplifies to:
\[ x + y = 3 \quad \text{(6)} \]

6. **Step 5: Substitute \(z = 3\) back into equation (5) to find \(x + 2y\).**
From equation (5):
\[ x + 2y = 10 - \frac{5(3)}{3} \]
Simplifying gives:
\[ x + 2y = 10 - 5 = 5 \quad \text{(7)} \]

7. **Step 6: Now solve equations (6) and (7) simultaneously.**
From (6):
\[ x + y = 3 \]
From (7):
\[ x + 2y = 5 \]
Subtract equation (6) from (7):
\[ (x + 2y) - (x + y) = 5 - 3 \]
This simplifies to:
\[ y = 2 \]

8. **Step 7: Substitute \(y = 2\) back into equation (6) to find \(x\).**
\[ x + 2 = 3 \quad \Rightarrow \quad x = 1 \]

9. **Final Solution:**
The values are:
\[ x = 1, \quad y = 2, \quad z = 3 \]