If `3x + 2y + 4z = 20, 5x + 7y - 9z = - 10` which of the following can be a set of values of x , y and z ?

Options

A. `x=2,y = - 1, z = 4`
B. `x = 4, y = 2 , z = 1`
C. `x = 1, y = 3 , z = 2`
D. `x = 2, y = 1 , z = 3`

Answer

To solve the problem, we need to determine which of the given options can satisfy the two equations:

1. \(3x + 2y + 4z = 20\)
2. \(5x + 7y - 9z = -10\)

Since we have three variables (x, y, z) and only two equations, we cannot find unique values for x, y, and z. Instead, we will check each option provided to see if they satisfy both equations.

### Step-by-Step Solution:

**Step 1: Check Option A**
- Let \(x = 2y - 1\) and \(z = 4\).
- Substitute into the first equation:
\[ 3(2y - 1) + 2y + 4(4) = 20 \]
\[ 6y - 3 + 2y + 16 = 20 \]
\[ 8y + 13 = 20 \]
\[ 8y = 7 \implies y = \frac{7}{8} \]
- Now substitute \(x\) and \(y\) into the second equation:
\[ 5(2y - 1) + 7y - 9(4) = -10 \]
\[ 10y - 5 + 7y - 36 = -10 \]
\[ 17y - 41 = -10 \]
\[ 17y = 31 \implies y = \frac{31}{17} \]
- Since the values of \(y\) do not match, **Option A is incorrect**.

**Step 2: Check Option B**
- Let \(x = 4\), \(y = 2\), and \(z = 1\).
- Substitute into the first equation:
\[ 3(4) + 2(2) + 4(1) = 20 \]
\[ 12 + 4 + 4 = 20 \]
- This satisfies the first equation.
- Now substitute into the second equation:
\[ 5(4) + 7(2) - 9(1) = -10 \]
\[ 20 + 14 - 9 = 25 \neq -10 \]
- Since the second equation is not satisfied, **Option B is incorrect**.

**Step 3: Check Option C**
- Let \(x = 1\), \(y = 3\), and \(z = 2\).
- Substitute into the first equation:
\[ 3(1) + 2(3) + 4(2) = 20 \]
\[ 3 + 6 + 8 = 17 \neq 20 \]
- Since the first equation is not satisfied, **Option C is incorrect**.

**Step 4: Check Option D**
- Let \(x = 2\), \(y = 1\), and \(z = 3\).
- Substitute into the first equation:
\[ 3(2) + 2(1) + 4(3) = 20 \]
\[ 6 + 2 + 12 = 20 \]
- This satisfies the first equation.
- Now substitute into the second equation:
\[ 5(2) + 7(1) - 9(3) = -10 \]
\[ 10 + 7 - 27 = -10 \]
- This satisfies the second equation as well.
- Since both equations are satisfied, **Option D is correct**.

### Final Answer:
The set of values that satisfies both equations is given by **Option D: \(x = 2\), \(y = 1\), \(z = 3\)**.