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:

\[ 3x – 5y = 20, \qquad 6x – 10y = 40 \]

Solution

Step 1: Write in Standard Form

\[ 3x – 5y – 20 = 0 \quad (1) \]

\[ 6x – 10y – 40 = 0 \quad (2) \]

Step 2: Compare Coefficients

From equations (1) and (2),

\[ a_1 = 3, \quad b_1 = -5, \quad c_1 = -20 \]

\[ a_2 = 6, \quad b_2 = -10, \quad c_2 = -40 \]

Step 3: Check Consistency Conditions

\[ \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}, \qquad \frac{b_1}{b_2} = \frac{-5}{-10} = \frac{1}{2}, \qquad \frac{c_1}{c_2} = \frac{-20}{-40} = \frac{1}{2} \]

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:

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

\[ \therefore \quad 3x – 5y = 20 \text{ and } 6x – 10y = 40 \text{ represent the same line.} \]