Question # To solve:\displaystyle{\left(\begin{matrix}{x}-{2}{y}={2}\\{2}{x}+{3}{y}={11}\\{y}-{4}{z}=-{7}\end{matrix}\right)}

Alternate coordinate systems
ANSWERED To solve:
$$\displaystyle{\left(\begin{matrix}{x}-{2}{y}={2}\\{2}{x}+{3}{y}={11}\\{y}-{4}{z}=-{7}\end{matrix}\right)}$$ 2020-10-22

Calculation:
We have to solve
$$\displaystyle{x}-{2}{y}={2}\ldots{\left({1}\right)}$$
$$\displaystyle{2}{x}+{3}{y}={11}\ldots{\left({2}\right)}$$
$$\displaystyle{y}-{4}{z}=-{7}\ldots{\left({3}\right)}$$
We have to choose equation 1 and equation 2 and eliminate the same variable x.
Multiply both sides of first equation by -2 and we have to add equation 1 and 2, we get
$$\displaystyle{\left({1}\right)}\times-{2}\to-{2}{x}+{4}{y}=-{4}$$
$$\displaystyle{\left({2}\right)}\to{2}{x}+{3}{y}={11}$$
$$\displaystyle{7}{y}={7}$$
$$\displaystyle{y}={1}$$
Now we have to choose a different pair of equations and eliminate the same variable.
Substitude $$\displaystyle{y}={1}$$ in equation 1, we get
$$\displaystyle{x}-{2}{y}={2}$$
$$\displaystyle{x}-{2}{\left({1}\right)}={2}$$
$$\displaystyle{x}-{2}={2}$$
$$\displaystyle{x}={2}+{2}$$
$$\displaystyle{x}={4}$$
Substitute $$\displaystyle{y}={1}$$ in equation 3, we get
$$\displaystyle{y}-{4}{z}=-{7}$$
$$\displaystyle{1}-{4}{z}=-{7}$$
$$\displaystyle-{4}{z}=-{7}-{1}$$
$$\displaystyle-{4}{z}=-{8}$$
Divide both sides by -4
$$\displaystyle\frac{{-{4}{z}}}{{-{4}}}=\frac{{-{8}}}{{-{4}}}$$
$$\displaystyle{z}={2}$$
The solution set for the system is {(4, 1, 2)}
To check:
Substitute $$\displaystyle{x}={4},{y}={1}$$ in equation 1
$$\displaystyle{x}-{2}{y}={2}$$
$$\displaystyle{4}-{2}{\left({1}\right)}={2}$$
$$\displaystyle{4}-{2}={2}$$
$$\displaystyle{2}={2}$$[True]
Substitute $$\displaystyle{x}={4},{y}={1}$$ in equation 2
$$\displaystyle{2}{x}+{3}{y}={11}$$
$$\displaystyle{2}{\left({4}\right)}+{3}{\left({1}\right)}={11}$$
$$\displaystyle{8}+{3}={11}$$
$$\displaystyle{11}={11}$$ [True]
Substitute $$\displaystyle{y}={1},{z}={2}$$ in equation 3
$$\displaystyle{y}-{4}{z}=-{7}$$
$$\displaystyle{1}-{4}{\left({2}\right)}=-{7}$$
$$\displaystyle{1}-{8}=-{7}$$
$$\displaystyle-{7}=-{7}$$ [True]
The solution set $$\displaystyle={\left\lbrace{\left({4},{1},{2}\right)}\right\rbrace}$$
Conclusion:
The solution set for the system is {(4, 1, 2)}