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 equations has infinitely many solutions:

\[ 2x + 3y = 7, \qquad 2ax + ay = 28 – by \]

Solution

Step 1: Write Both Equations in Standard Form

\[ 2x + 3y – 7 = 0 \quad (1) \]

\[ 2ax + ay + by – 28 = 0 \]

\[ 2ax + (a+b)y – 28 = 0 \quad (2) \]

Step 2: Identify Coefficients

From equations (1) and (2),

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

\[ a_2 = 2a, \quad b_2 = a+b, \quad c_2 = -28 \]

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

\[ \frac{c_1}{c_2} = \frac{-7}{-28} = \frac{1}{4} \]

So,

\[ \frac{2}{2a} = \frac{1}{4} \quad \text{and} \quad \frac{3}{a+b} = \frac{1}{4} \]

Step 5: Find the Value of a

\[ \frac{1}{a} = \frac{1}{4} \]

\[ a = 4 \]

Step 6: Find the Value of b

\[ a + b = 12 \]

\[ 4 + b = 12 \]

\[ b = 8 \]

Conclusion

The given system of equations has infinitely many solutions for:

\[ \boxed{a = 4, \quad b = 8} \]

\[ \therefore \quad 2x + 3y = 7 \text{ and } 8x + 12y = 28 \text{ represent the same line.} \]