the two equations `x - 4y = 0 and 4 x + 3y = 19` have a solution (a, b) . Find the value of ` ((ab)/( a + 4 b))`

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. 1
D. 23

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) from the given equations and then calculate the expression \( \frac{ab}{a + 4b} \).

### Step-by-Step Solution:

1. **Write the equations**:
The two equations given are:
\[ x - 4y = 0 \quad \text{(1)} \]
\[ 4x + 3y = 19 \quad \text{(2)} \]

2. **Express \( x \) in terms of \( y \)**:
From equation (1), we can express \( x \) in terms of \( y \):
\[ x = 4y \quad \text{(3)} \]

3. **Substitute \( x \) in equation (2)**:
Substitute equation (3) into equation (2):
\[ 4(4y) + 3y = 19 \]
Simplifying this gives:
\[ 16y + 3y = 19 \]
\[ 19y = 19 \]

4. **Solve for \( y \)**:
Dividing both sides by 19:
\[ y = 1 \quad \text{(4)} \]

5. **Find \( x \)**:
Now substitute \( y \) back into equation (3) to find \( x \):
\[ x = 4(1) = 4 \quad \text{(5)} \]

6. **Identify \( a \) and \( b \)**:
From equations (4) and (5), we have:
\[ a = x = 4 \quad \text{and} \quad b = y = 1 \]

7. **Calculate \( \frac{ab}{a + 4b} \)**:
Now substitute \( a \) and \( b \) into the expression:
\[ \frac{ab}{a + 4b} = \frac{(4)(1)}{4 + 4(1)} \]
Simplifying this gives:
\[ \frac{4}{4 + 4} = \frac{4}{8} = \frac{1}{2} \]

### Final Answer:
The value of \( \frac{ab}{a + 4b} \) is \( \frac{1}{2} \).

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