2x1 + x2 = 3
5x1 + bx2 = 7.5
The system of linear equations in two variables shown above will have infinite solutions, if and
only if, b is equal to
2.5
Official Answer Key:
2.5 to 2.5
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2015 Q.34)**
**1. Governing Concept & Definition**:
- **Topic**: Linear Algebra & Consistency of Linear Equations
- For a system of two equations:
$a_1 x_1 + b_1 x_2 = c_1$
$a_2 x_1 + b_2 x_2 = c_2$
to have infinitely many solutions, the coefficient ratios must be identical:
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
**2. Detailed Step-by-Step Calculation**:
- **Given Equations**:
1. $2x_1 + x_2 = 3$
2. $5x_1 + b x_2 = 7.5$
- **Ratios**:
$\frac{a_1}{a_2} = \frac{2}{5} = 0.4$
$\frac{c_1}{c_2} = \frac{3}{7.5} = \frac{1}{2.5} = 0.4$
- For infinite solutions:
$\frac{1}{b} = 0.4 \implies b = \frac{1}{0.4} = 2.5$
**3. Final Answer**:
- **Calculated Value**: **2.5**
- **Official GATE Accepted Range**: **2.5 to 2.5**
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