If ` x + 3y - ( 2 z)/( 4) = 6 , x + (2)/(3) ( 2 y + 3 z) = 33 and (1)/(7) ( x + y + z) + 2 z = 9 ` then what is the value of ` 46 x + 131y `

Options

A. 414
B. 364
C. 384
D. 464

Answer

To solve the given system of equations and find the value of \(46x + 131y\), we will follow these steps:

### Step 1: Write down the equations
The equations provided are:
1. \( x + 3y - \frac{2z}{4} = 6 \)
2. \( x + \frac{2}{3}(2y + 3z) = 33 \)
3. \( \frac{1}{7}(x + y + z) + 2z = 9 \)

### Step 2: Simplify the equations
Let's simplify each equation:

**Equation 1:**
\[ x + 3y - \frac{z}{2} = 6 \quad \text{(Multiply through by 2 to eliminate the fraction)} \]
\[ 2x + 6y - z = 12 \quad \text{(Equation 1 simplified)} \]

**Equation 2:**
\[ x + \frac{2}{3}(2y + 3z) = 33 \quad \text{(Distributing the \(\frac{2}{3}\))} \]
\[ x + \frac{4y}{3} + 2z = 33 \quad \text{(Multiply through by 3 to eliminate the fraction)} \]
\[ 3x + 4y + 6z = 99 \quad \text{(Equation 2 simplified)} \]

**Equation 3:**
\[ \frac{1}{7}(x + y + z) + 2z = 9 \quad \text{(Multiply through by 7 to eliminate the fraction)} \]
\[ x + y + z + 14z = 63 \]
\[ x + y + 15z = 63 \quad \text{(Equation 3 simplified)} \]

### Step 3: Solve the system of equations
We now have the following simplified equations:
1. \( 2x + 6y - z = 12 \) (Equation 1)
2. \( 3x + 4y + 6z = 99 \) (Equation 2)
3. \( x + y + 15z = 63 \) (Equation 3)

Next, we can express \(z\) in terms of \(x\) and \(y\) using Equation 1:
\[ z = 2x + 6y - 12 \]

Now, substitute \(z\) into Equations 2 and 3.

**Substituting into Equation 2:**
\[ 3x + 4y + 6(2x + 6y - 12) = 99 \]
\[ 3x + 4y + 12x + 36y - 72 = 99 \]
\[ 15x + 40y - 72 = 99 \]
\[ 15x + 40y = 171 \quad \text{(Equation 4)} \]

**Substituting into Equation 3:**
\[ x + y + 15(2x + 6y - 12) = 63 \]
\[ x + y + 30x + 90y - 180 = 63 \]
\[ 31x + 91y - 180 = 63 \]
\[ 31x + 91y = 243 \quad \text{(Equation 5)} \]

### Step 4: Solve Equations 4 and 5
Now we solve the system formed by Equations 4 and 5:
1. \( 15x + 40y = 171 \)
2. \( 31x + 91y = 243 \)

We can multiply Equation 4 by 31 and Equation 5 by 15 to eliminate \(x\):
\[ 31(15x + 40y) = 31(171) \quad \Rightarrow \quad 465x + 1240y = 5281 \]
\[ 15(31x + 91y) = 15(243) \quad \Rightarrow \quad 465x + 1365y = 3645 \]

Now, subtract the first modified equation from the second:
\[ (465x + 1365y) - (465x + 1240y) = 3645 - 5281 \]
\[ 125y = -1636 \]
\[ y = -\frac{1636}{125} = -13.088 \]

### Step 5: Substitute \(y\) back to find \(x\) and \(z\)
Substituting \(y\) back into Equation 4:
\[ 15x + 40(-13.088) = 171 \]
\[ 15x - 523.52 = 171 \]
\[ 15x = 694.52 \]
\[ x = \frac{694.52}{15} = 46.297 \]

Now substitute \(x\) and \(y\) back to find \(z\):
\[ z = 2(46.297) + 6(-13.088) - 12 \]
\[ z = 92.594 - 78.528 - 12 = 2.066 \]

### Step 6: Calculate \(46x + 131y\)
Now we can find \(46x + 131y\):
\[ 46(46.297) + 131(-13.088) = 2139.662 - 1715.528 = 424.134 \]

### Final Answer
The value of \(46x + 131y\) is approximately \(424.134\).