Three UnknownsSolve these sets of three simultaneous, linear equations to find the values of the variables 
This is level 3: An awful heap of tedious equations generated by AI where the solutions are vulgar fractions (not recommended). You will be awarded a trophy if you get at least 15 answers correct and you do this activity online.
Type fractions greater than one as improper fractions rather than mixed numbers. Use the forward slash "/" symbol in your fractional answers.
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❎Two unknowns  You really should start here before taking on three unknowns.
Level 1  Standard set of questions with the equations set out in a familiar way
Level 2  A mixed up collection of equations to challenge the high achiever
Level 3  An awful heap of tedious equations generated by AI where the solutions are vulgar fractions (not recommended)
More on this topic including lesson Starters, visual aids, investigations and selfmarking exercises.
The video above is from Corbett Maths.
Example using matrix row reduction:
Solve the following system of equations:
$$x+y+z=3$$ $$2xy+z=0$$ $$x2yz=3$$We begin by writing the augmented matrix for the system of equations:
$$\left[\begin{array}{cccc}1 & 1 & 1 & 3\\2 & 1 & 1 & 0\\1 & 2 & 1 & 3\end{array}\right]$$We now perform row operations to transform this matrix into echelon form:
\(R2 = R2  2 \times R1\)
$$\left[\begin{array}{cccc}1 & 1 & 1 & 3\\0 & 3 & 1 & 6\\1 & 2 & 1 & 3\end{array}\right]$$\(R3 = R3  R1\)
$$\left[\begin{array}{cccc}1 & 1 & 1 & 3\\0 & 3 & 1 & 6\\0 & 3 & 2 & 6\end{array}\right]$$\(R3 = R3  R2\)
$$\left[\begin{array}{cccc}1 & 1 & 1 & 3\\0 & 3 & 1 & 6\\0 & 0 & 1 & 0\end{array}\right]$$The augmented matrix is now in the form:
$$\left[\begin{array}{cccc}a & b & c & d\\0 & e & f & g\\0 & 0 & h & i\end{array}\right]$$If \( h \neq 0\) there is a unique solution.
If \( h = 0 \text{ and } i \neq 0 \) there is no solution
If \( h = 0 \text{ and } i = 0 \) there are infinitely many solutions (let \(z=t\)).
In the example above ...
from \(R3\) it can be seen that \( z=0 \)
from \(R2\) it can be seen that \( y=2 \)
from \(R1\) it can be seen that \( x=1 \)
The solutions are \( x=1,y=2,z=0 \).
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