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:

\[ x – 2y = 8, \qquad 5x – 10y = 10 \]

Solution

Step 1: Write in Standard Form

\[ x – 2y – 8 = 0 \quad (1) \]

\[ 5x – 10y – 10 = 0 \quad (2) \]

Step 2: Compare Coefficients

From equations (1) and (2),

\[ a_1 = 1, \quad b_1 = -2, \quad c_1 = -8 \]

\[ a_2 = 5, \quad b_2 = -10, \quad c_2 = -10 \]

Step 3: Check Consistency Conditions

\[ \frac{a_1}{a_2} = \frac{1}{5}, \qquad \frac{b_1}{b_2} = \frac{-2}{-10} = \frac{1}{5}, \qquad \frac{c_1}{c_2} = \frac{-8}{-10} = \frac{4}{5} \]

Step 4: Analyze the Ratios

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

Hence, the given pair of linear equations is inconsistent.

Conclusion

The given system of equations has:

\[ \boxed{\text{No solution}} \]

\[ \therefore \quad x – 2y = 8 \text{ and } 5x – 10y = 10 \text{ represent parallel lines.} \]