Three equations (I), (II) and (III) are given in each question. On the basis of these equations you have to decide the relation between 'x' , 'y' and 'z' and give answer
I. ` 7 x + 6 y + 4 z = 122`
II. ` 4 x + 5 y + 3 z = 88`
III. ` 9 x + 2 y + z = 78`

Options

A. If ` x lt y = z `
B. If ` x le y lt z `
C. If ` x lt y gt z `
D. If ` x = y gt z `

Answer

To solve the given equations and find the relationship between \( x \), \( y \), and \( z \), we will follow a systematic approach.

### Step 1: Write down the equations
We have the following equations:
1. \( 7x + 6y + 4z = 122 \) (Equation I)
2. \( 4x + 5y + 3z = 88 \) (Equation II)
3. \( 9x + 2y + z = 78 \) (Equation III)

### Step 2: Manipulate the equations
We will manipulate these equations to eliminate variables and find the values of \( x \), \( y \), and \( z \).

#### Step 2.1: Multiply Equation III by 3
Multiply Equation III by 3:
\[ 3(9x + 2y + z) = 3(78) \]
This gives us:
\[ 27x + 6y + 3z = 234 \quad \text{(Equation IV)} \]

#### Step 2.2: Subtract Equation II from Equation IV
Now, we will subtract Equation II from Equation IV:
\[ (27x + 6y + 3z) - (4x + 5y + 3z) = 234 - 88 \]
This simplifies to:
\[ 23x + y = 146 \quad \text{(Equation V)} \]

### Step 3: Manipulate Equations I and III
#### Step 3.1: Multiply Equation III by 4
Multiply Equation III by 4:
\[ 4(9x + 2y + z) = 4(78) \]
This gives us:
\[ 36x + 8y + 4z = 312 \quad \text{(Equation VI)} \]

#### Step 3.2: Subtract Equation I from Equation VI
Now, we will subtract Equation I from Equation VI:
\[ (36x + 8y + 4z) - (7x + 6y + 4z) = 312 - 122 \]
This simplifies to:
\[ 29x + 2y = 190 \quad \text{(Equation VII)} \]

### Step 4: Solve Equations V and VII
Now we have two new equations:
1. \( 23x + y = 146 \) (Equation V)
2. \( 29x + 2y = 190 \) (Equation VII)

#### Step 4.1: Multiply Equation V by 2
Multiply Equation V by 2:
\[ 2(23x + y) = 2(146) \]
This gives us:
\[ 46x + 2y = 292 \quad \text{(Equation VIII)} \]

#### Step 4.2: Subtract Equation VII from Equation VIII
Now, we will subtract Equation VII from Equation VIII:
\[ (46x + 2y) - (29x + 2y) = 292 - 190 \]
This simplifies to:
\[ 17x = 102 \]
Thus, we find:
\[ x = 6 \]

### Step 5: Find \( y \)
Substituting \( x = 6 \) back into Equation V:
\[ 23(6) + y = 146 \]
This simplifies to:
\[ 138 + y = 146 \implies y = 8 \]

### Step 6: Find \( z \)
Now, substituting \( x = 6 \) and \( y = 8 \) back into Equation III:
\[ 9(6) + 2(8) + z = 78 \]
This simplifies to:
\[ 54 + 16 + z = 78 \implies z = 78 - 70 = 8 \]

### Conclusion: Values of \( x \), \( y \), and \( z \)
We have found:
- \( x = 6 \)
- \( y = 8 \)
- \( z = 8 \)

### Step 7: Determine the relationship
Now we can determine the relationship between \( x \), \( y \), and \( z \):
- \( x < y \)
- \( x < z \)
- \( y = z \)

### Final Answer
The relationship is \( x < y = z \).

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