Multiple choice

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

  1. 2, 7

  2. 3, 8

  3. 8, 3

  4. 7, 2
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A Correct answer
Explanation

For infinite solutions, the augmented matrix must have a rank less than the number of variables. Solving the system x+y+z=5 and x+3y+3z=9 gives 2y+2z=4, or y+z=2. Substituting into the third equation x+2y+alpha*z=beta, we need the equations to be consistent, which occurs when alpha=2 and beta=7.

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