I have just learned about Gaussian Elimination and I decided to try an example question.
I was trying to solve a question and I realised later that I had copied the question wrong but I still decided to proceed and solve the system of linear equations.
However, when I checked my answer online using an online calculator it gave different values meaning my solution was wrong. I've checked my working out and I can't seem to figure out where I've gone wrong. I'd appreciate if someone could help me figure out my mistake.
I've used an online tool to convert my writing into LATEX but it's kind of messed up. I have no idea how to write in LATEX. As a result, I have attached images as well:
Also, I'd appreciate any suggestions on how to improve my work.
(Next time, I'm just going to pivot on the 3 to avoid fractions.)
Here it is:
Q) $$ \begin{aligned} & 3 x-2 y-4 z=3 \\ & 2 x+3 y+3 z=15 \\ & 5 x-3 y+z=14 \\ & {\left[\begin{array}{rrr|r} 3^* & -2 & -4 & 3 \\ 2 & 3 & 3 & 15 \\ 5 & -3 & 1 & 14 \end{array}\right] \quad R_1 \times \frac{1}{3}=R_1} \end{aligned} $$ $$ \begin{aligned} & {\left[\begin{array}{ccc|c} 1 & -\frac{2}{3} & -\frac{4}{3} & 1 \\ 2 & 3 & 3 & 15 \\ 5 & -3 & 1 & 14 \end{array}\right] R_2-2 R_1 \rightarrow R_2} \\ & {\left[\begin{array}{ccc|c} 1 & -\frac{2}{3} & -4 / 3 & 1 \\ 0 & 3 / 3 & 17 / 3 & 13 \\ 5 & -3 & 1 & 14 \end{array}\right] R_3-5 R_1 \rightarrow R_3} \end{aligned} $$ $$ \left[\begin{array}{ccc|c} 1 & -2 / 3 & -4 / 3 & 1 \\ 0 & 13 / 3 & 17 / 3 & 13 \\ 0 & 1 / 3 & 23 / 3 & 9 \end{array}\right] \text { } $$
\begin{aligned} & {\left[\begin{array}{ccc|c} 1 & -2 / 3 & -4 / 3 & 1 \\ 0 & 1 & 17 / 13 & 3 \\ 0 & 1 / 3 & 23 / 3 & 9 \end{array}\right] \quad \begin{array}{l} R_2 \times 3 / 13 \\ R_3-1 / 3 R_2 \rightarrow R_3 \end{array}} \\ & {\left[\begin{array}{ccc|c} 1 & 0 & -6 / 13 & 1 \\ 0 & 1 & 17 / 13 & 3 \\ 0 & 0 & 94 / 13 & 11 \end{array}\right] \quad R_1+2 / 3 R_2 \rightarrow R_1} \\ & {\left[\begin{array}{ccc|c|} 1 & 0 & -6 / 1 & 1 \\ 0 & 1 & 17 / 13 & 3 \\ 0 & 0 & 1 & 143 / 94 \end{array}\right] \quad \begin{array}{l} R_1+\frac{6 R_3}{13} \rightarrow R_1 \\ R_2-\frac{17 R_3}{3} \rightarrow R_2 \end{array}} \\ & {\left[\begin{array} { l l l | l } { 1 } & { 0 } & { 0 } & { 8 0 / 4 7 } \\ { 0 } & { 1 } & { 0 } & { 9 5 / 9 4 } \\ { 0 } & { 0 } & { 1 } & { 1 4 3 / 9 4 } \end{array} \quad \left[x=\frac{165}{47} \quad y=\frac{73}{47} z=\frac{52}{47}\right.\right.} \\ & \end{aligned}
You have done a good work but may be you have missed something near this step (verify again):
$$R1+6\:\frac{R3}{13}\:->\:R1$$ $$R2\:-17\:\frac{R3}{3}\:->\:R2$$
Here is my try, I could not keep using fractions till the end, because of cell 3,3 near the end.