Science:Math Exam Resources/Courses/MATH221/April 2013/Question 02 (d)/Solution 1

If we write the system of equations as an extended matrix, it will be exactly the matrix A. Thus, thinking of A as an extended matrix, we know it represents 4 equations in 4 variables. Row operations lead to an equivalent system of equations, with the same solutions. Hence we can instead choose the extended matrix B, even better, we can further use the reduced matrix we used in part (b), i.e. the matrix:

$\left[{\begin{array}{ccccc}1&0&7&0&-39\\0&1&-3&0&31\\0&0&0&1&-7\\0&0&0&0&0\end{array}}\right]$

to determine the answer. From the above matrix, we can read the equations:

${\begin{aligned}&x+7z&=-39\\&y-3z&=31\\&w&=-7\end{aligned}}$

From this, we can quickly see that $w=0$ leads to no solution (i.e. it would lead to saying 0 = -7).

If z=0, then we get the solution

$\displaystyle x=-39,\quad y=31,\quad w=-7$