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
asked 2020-10-21

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

Answers (1)

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)}

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