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:

\[ 2x + 3y = 7, \qquad (a-b)x + (a+b)y = 3a + b – 2 \]

Solution

Step 1: Write the Equations in Standard Form

\[ 2x + 3y – 7 = 0 \quad (1) \]

\[ (a-b)x + (a+b)y – (3a+b-2) = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

\[ a_1 = 2, \quad b_1 = 3, \quad c_1 = -7 \]

\[ a_2 = a-b, \quad b_2 = a+b, \quad c_2 = -(3a+b-2) \]

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

Equating the first two ratios:

\[ \frac{2}{a-b} = \frac{3}{a+b} \]

\[ 2(a+b) = 3(a-b) \]

\[ 2a + 2b = 3a – 3b \]

\[ a = 5b \quad (i) \]

Now equate the first and third ratios:

\[ \frac{2}{a-b} = \frac{7}{3a+b-2} \]

\[ 2(3a+b-2) = 7(a-b) \]

\[ 6a + 2b – 4 = 7a – 7b \]

\[ a – 9b = 4 \quad (ii) \]

Step 5: Solve the System

From (i): \(a = 5b\)

Substitute in (ii):

\[ 5b – 9b = 4 \]

\[ -4b = 4 \]

\[ b = -1 \]

\[ a = 5(-1) = -5 \]

Conclusion

The given system of equations has infinitely many solutions for:

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

\[ \therefore \quad 2x + 3y = 7 \text{ and } -4x – 6y = -14 \text{ represent the same line.} \]