Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 02 (d)

MATH152 April 2016

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Question B 02 (d)
In the system below, $x$, $y$, and $z$ are variables, and $a$ and $b$ are constants. ${\begin{array}{ccccccc}x&+&y&+&z&=&5\\x&~&~&+&z&=&1\\ax&~&~&+&z&=&b\end{array}}$ (d) For what value or values of $a$ and $b$ (if any) does the system have an infinite number of solutions?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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Hint
The system has infinitely many solutions if a row echelon form of the augmented matrix has more variables than non-zero rows.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By (b), a row echelon form of the augmented matrix is $\left({\begin{array}{ccc|c}1&1&1&5\\0&-1&0&-4\\0&0&1-a&b-a\end{array}}\right).$ The system has infinitely many solutions if the row echelon form has more variables than non-zero rows. We have 3 variables $x,y,z$. Note that first two rows cannot be zero rows because of the nonzero pivots. In order for the system to have infinitely many solutions, the third row must be a zero row. Therefore, we need $1-a=0$ and $b-a=0$, so that the answer is $\color {blue}a=b=1$.