# Science:Math Exam Resources/Courses/MATH152/April 2022/Question B1 (a)/Solution 1

The linear system described in the hint is given by the following augmented matrix:

${\displaystyle \left[{\begin{array}{cccc|c}1&2&1&3&11\\3&6&3&9&33\\1&4&3&7&21\\2&1&3&8&23\\0&5&1&2&H\end{array}}\right]}$

We can solve the linear system by applying row operations (we use the notation ${\displaystyle L_{i,j}(m)}$ for the operation "add ${\displaystyle m}$ times the ${\displaystyle j^{\mathrm {th} }}$ row to the ${\displaystyle i^{\mathrm {th} }}$ row" and ${\displaystyle D_{i}(m)}$ for "multiply the ${\displaystyle i^{\mathrm {th} }}$ row by ${\displaystyle m}$"):

{\displaystyle {\begin{aligned}\left[{\begin{array}{cccc|c}1&2&1&3&11\\3&6&3&9&33\\1&4&3&7&21\\2&1&3&8&23\\0&5&1&2&H\end{array}}\right]\xrightarrow {L_{2,1}(-3),\ L_{3,1}(-1),\ L_{4,1}(-2)} &\left[{\begin{array}{cccc|c}1&2&1&3&11\\0&0&0&0&0\\0&2&2&4&10\\0&-3&1&2&1\\0&5&1&2&H\end{array}}\right]\\\xrightarrow {D_{3}(1/2),\ L_{1,3}(-2),\ L_{4,3}(3),\ L_{5,3}(-5)} &\left[{\begin{array}{cccc|c}1&0&-1&-1&1\\0&0&0&0&0\\0&1&1&2&5\\0&0&4&8&16\\0&0&-4&-8&H-25\end{array}}\right]\\\xrightarrow {D_{4}(1/4),\ L_{1,4}(1),\ L_{3,4}(-1),\ L_{5,4}(1)} &\left[{\begin{array}{cccc|c}1&0&0&1&5\\0&0&0&0&0\\0&1&0&0&1\\0&0&1&2&4\\0&0&0&0&H-9\end{array}}\right]\end{aligned}}}

From the last row, we see that ${\displaystyle H=9}$.