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35 votes
35 votes

Consider the following system of equations:  

  • $3x + 2y = 1 $
  • $4x + 7z = 1 $
  • $x + y + z = 3$
  • $x - 2y + 7z = 0$

The number of solutions for this system is ______________

6 Answers

Best answer
45 votes
45 votes
Since, equation $(2)$ - equation $(1)$ produces equation $(4)$, we have $3$ independent equations in $3$ variables, hence unique solution.

So, answer is $1.$
edited by
49 votes
49 votes

sorry for my handwriting!

16 votes
16 votes
Add first two equations and you will get $7x+2y+7z=2$

remaining equations are $x+y+z=3$ and $x-2y+7z=0$

My augmented matrix

$\left[\begin{array}{ccc|c} 1&1&1&3 \\ 7&2&7&2 \\ 1&-2&7&0 \\ \end{array} \right ]$

do $R_2-7R_1 \rightarrow R_2$ and $R_3-R_1 \rightarrow R_3$

$\left[\begin{array}{ccc|c} 1&1&1&3 \\ 0&-5&0&-19 \\ 0&-3&6&-3 \\ \end{array} \right ]$

do $5R_3 - 3R_2 \rightarrow R_3$

$\left[\begin{array}{ccc|c} 1&1&1&3 \\ 0&-5&0&-19 \\ 0&0&30&42 \\ \end{array} \right ]$

your Augmented matrix has same rank as the coefficient matrix=3=number of unknowns, so only 1 solution possible.
0 votes
0 votes
rank(Augmented Matrix) = rank(Matrix) = no of unknowns. Hence it has a unique solution
Answer:

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