Finding Values of a and b for Infinite Solutions

Video Explanation

Question

If the system of equations \(2x + 3y = 7\) and \((a+b)x + (2a-b)y = 21\) has infinitely many solutions, find \(a\) and \(b\).

Solution

Step 1: Write in Standard Form

\[ 2x + 3y – 7 = 0 \]

\[ (a+b)x + (2a-b)y – 21 = 0 \]

Step 2: Apply Condition for Infinite Solutions

For infinitely many solutions:

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

\[ \frac{2}{a+b} = \frac{3}{2a-b} = \frac{-7}{-21} \]

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

Step 3: Form Equations

\[ \frac{2}{a+b} = \frac{1}{3} \Rightarrow a + b = 6 \]

\[ \frac{3}{2a-b} = \frac{1}{3} \Rightarrow 2a – b = 9 \]

Step 4: Solve System

Add equations:

\[ (a + b) + (2a – b) = 6 + 9 \]

\[ 3a = 15 \Rightarrow a = 5 \]

Substitute into \(a + b = 6\):

\[ 5 + b = 6 \Rightarrow b = 1 \]

Final Answer

\[ a = 5,\quad b = 1 \]