Consistency of a Pair of Linear Equations

Video Explanation

Question

Determine whether the following system of equations has a unique solution, no solution or infinitely many solutions. If it has a unique solution, find it:

\[ 2x + y = 5, \qquad 4x + 2y = 10 \]

Solution

Step 1: Write in Standard Form

\[ 2x + y – 5 = 0 \quad (1) \]

\[ 4x + 2y – 10 = 0 \quad (2) \]

Step 2: Compare Coefficients

From equations (1) and (2),

\[ a_1 = 2, \quad b_1 = 1, \quad c_1 = -5 \]

\[ a_2 = 4, \quad b_2 = 2, \quad c_2 = -10 \]

Step 3: Check Consistency Conditions

\[ \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}, \qquad \frac{b_1}{b_2} = \frac{1}{2}, \qquad \frac{c_1}{c_2} = \frac{-5}{-10} = \frac{1}{2} \]

Step 4: Analyze the Ratios

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Hence, the given pair of linear equations is consistent and dependent.

Conclusion

The given system of equations has:

\[ \boxed{\text{Infinitely many solutions}} \]

\[ \therefore \quad 2x + y = 5 \text{ and } 4x + 2y = 10 \text{ represent the same line.} \]