Infinitely Many Solutions of a Pair of Linear Equations

Video Explanation

Question

Find the values of \(a\) and \(b\) for which the following system of linear equations has infinitely many solutions:

\[ (a-1)x + 3y = 2, \qquad 6x + (1-2b)y = 6 \]

Solution

Step 1: Write in Standard Form

\[ (a-1)x + 3y – 2 = 0 \quad (1) \]

\[ 6x + (1-2b)y – 6 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = a-1, \quad b_1 = 3, \quad c_1 = -2 \]

\[ a_2 = 6, \quad b_2 = 1-2b, \quad c_2 = -6 \]

Step 3: Condition for Infinitely Many Solutions

A pair of linear equations has infinitely many solutions if

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

Step 4: Apply the Condition

\[ \frac{c_1}{c_2} = \frac{-2}{-6} = \frac{1}{3} \]

So,

\[ \frac{a-1}{6} = \frac{1}{3} \quad \text{and} \quad \frac{3}{1-2b} = \frac{1}{3} \]

Step 5: Find the Value of a

\[ a – 1 = 2 \]

\[ a = 3 \]

Step 6: Find the Value of b

\[ 9 = 1 – 2b \]

\[ 2b = -8 \]

\[ b = -4 \]

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{a = 3, \quad b = -4} \]

\[ \therefore \quad 2x + 3y = 2 \text{ and } 6x + 9y = 6 \text{ represent the same line.} \]