Which of the following system of equations has unique solutions

Options

A. `3x + 4y = 11, 6x + 8y = 15`
B. `x + 2y = 3, 2x + 4y = 7`
C. `x + 2y = 3, 3x + 4y = 7`
D. `4x + 3y = 5, 4x - 3y = 5`

Answer

To determine which of the given systems of equations has a unique solution, we need to apply the condition for unique solutions in a system of linear equations. The condition states that for the system of equations:

1. \( a_1x + b_1y + c_1 = 0 \)
2. \( a_2x + b_2y + c_2 = 0 \)

The system has a unique solution if:

\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

Let's analyze each option step by step.

### Step 1: Analyze Option 1
- Given: \( a_1 = 3, b_1 = 4 \) and \( a_2 = 6, b_2 = 8 \)
- Calculate \( \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2} \)
- Calculate \( \frac{b_1}{b_2} = \frac{4}{8} = \frac{1}{2} \)
- Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \), this option does not satisfy the condition for unique solutions.

### Step 2: Analyze Option 2
- Given: \( a_1 = 1, b_1 = 2 \) and \( a_2 = 2, b_2 = 4 \)
- Calculate \( \frac{a_1}{a_2} = \frac{1}{2} \)
- Calculate \( \frac{b_1}{b_2} = \frac{2}{4} = \frac{1}{2} \)
- Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \), this option also does not satisfy the condition for unique solutions.

### Step 3: Analyze Option 3
- Given: \( a_1 = 1, b_1 = 2 \) and \( a_2 = 3, b_2 = 6 \)
- Calculate \( \frac{a_1}{a_2} = \frac{1}{3} \)
- Calculate \( \frac{b_1}{b_2} = \frac{2}{6} = \frac{1}{3} \)
- Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \), this option does not satisfy the condition for unique solutions.

### Step 4: Analyze Option 4
- Given: \( a_1 = 4, b_1 = 3 \) and \( a_2 = 4, b_2 = -3 \)
- Calculate \( \frac{a_1}{a_2} = \frac{4}{4} = 1 \)
- Calculate \( \frac{b_1}{b_2} = \frac{3}{-3} = -1 \)
- Since \( 1 \neq -1 \), this option satisfies the condition for unique solutions.

### Conclusion
The system of equations that has a unique solution is **Option 4**.

---