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:

\[ 2x – (2a+5)y = 5, \qquad (2b+1)x – 9y = 15 \]

Solution

Step 1: Write in Standard Form

\[ 2x – (2a+5)y – 5 = 0 \quad (1) \]

\[ (2b+1)x – 9y – 15 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = 2, \quad b_1 = -(2a+5), \quad c_1 = -5 \]

\[ a_2 = 2b+1, \quad b_2 = -9, \quad c_2 = -15 \]

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

\[ \frac{c_1}{c_2} = \frac{-5}{-15} = \frac{1}{3} \]

So,

\[ \frac{2}{2b+1} = \frac{1}{3} \quad \text{and} \quad \frac{2a+5}{9} = \frac{1}{3} \]

Step 5: Find the Value of b

\[ 6 = 2b + 1 \]

\[ 2b = 5 \]

\[ b = \frac{5}{2} \]

Step 6: Find the Value of a

\[ 2a + 5 = 3 \]

\[ 2a = -2 \]

\[ a = -1 \]

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{a = -1, \quad b = \frac{5}{2}} \]

\[ \therefore \quad 2x – 3y = 5 \text{ and } 6x – 9y = 15 \text{ represent the same line.} \]