Finding Relation Between a and b for Infinite Solutions

Question

If the system of equations \(2x + 3y = 7\) and \(2ax + (a+b)y = 28\) has infinitely many solutions, find the relation between \(a\) and \(b\).

Solution

Step 1: Write in Standard Form

\[ 2x + 3y – 7 = 0 \]

\[ 2ax + (a+b)y – 28 = 0 \]

Step 2: Apply Condition for Infinite Solutions

For infinitely many solutions:

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

\[ \frac{2}{2a} = \frac{3}{a+b} = \frac{-7}{-28} \]

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

Step 3: Solve

\[ \frac{2}{2a} = \frac{1}{4} \Rightarrow \frac{1}{a} = \frac{1}{4} \Rightarrow a = 4 \]

\[ \frac{3}{a+b} = \frac{1}{4} \Rightarrow a + b = 12 \]

Substitute \(a = 4\):

\[ 4 + b = 12 \Rightarrow b = 8 \]

Step 4: Required Relation

\[ b = 2a \]

Final Answer

\[ \text{Required relation: } b = 2a \]