Question

For what values of a and b the following simultaneous equations have an infinite number of solutions? $x + y + z = 5, x + 3y + 3z = 9$, and $x + 2y + \alpha z = \beta$.

• A. 2, 7
• B. 3, 8
• C. 8, 3
• D. 7, 2

Hint:

Infinite solutions occur when row reduction results in a row of zeros $(0,0,0 | 0)$.

Answer 1

The Correct Option is A

Step 1: Concept
A system has infinite solutions if the rank of the coefficient matrix equals the rank of the augmented matrix, but is less than the number of variables.

Step 2: Meaning
Set up augmented matrix: $\left[\begin{matrix}1& 1& 1&|& 5
1& 3& 3&|& 9
1& 2&\alpha&|&\beta\end{matrix}\right]$. Row operations: $R_2 \to R_2 - R_1, R_3 \to R_3 - R_1$: $\left[\begin{matrix}1& 1& 1&|& 5
0& 2& 2&|& 4
0& 1&\alpha-1&|&\beta-5\end{matrix}\right]$.

Step 3: Analysis
Further simplify $R_2$ by $1/2$: $\left[\begin{matrix}1& 1& 1&|& 5
0& 1& 1&|& 2
0& 1&\alpha-1&|&\beta-5\end{matrix}\right]$. $R_3 \to R_3 - R_2$: $\left[\begin{matrix}1& 1& 1&|& 5
0& 1& 1&|& 2
0& 0&\alpha-2&|&\beta-7\end{matrix}\right]$.

Step 4: Conclusion
For infinite solutions, the last row must be zero: $\alpha - 2 = 0 \Rightarrow \alpha = 2$ and $\beta - 7 = 0 \Rightarrow \beta = 7$. Final Answer: (A)