Guzman Guerrero
07/11/2023 · Primary School
b) Solve the set of equations using matrix method \[ \begin{array}{c}x_{1}+2 x_{2}+x_{3}=4 \\ 3 x_{1}-4 x_{2}-2 x_{3}=2 \\ 5 x_{1}+3 x_{2}+5 x_{3}=-1\end{array} \]
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Solve the system of equations \( x_{1}+2x_{2}+x_{3}=4;3x_{1}-4x_{2}-2x_{3}=2;5x_{1}+3x_{2}+5x_{3}=-1 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}x_{1}+2x_{2}+x_{3}=4\\3x_{1}-4x_{2}-2x_{3}=2\\5x_{1}+3x_{2}+5x_{3}=-1\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}x_{1}=4-2x_{2}-x_{3}\\3x_{1}-4x_{2}-2x_{3}=2\\5x_{1}+3x_{2}+5x_{3}=-1\end{array}\right.\)
- step2: Substitute the value of \(x_{1}:\)
\(\left\{ \begin{array}{l}3\left(4-2x_{2}-x_{3}\right)-4x_{2}-2x_{3}=2\\5\left(4-2x_{2}-x_{3}\right)+3x_{2}+5x_{3}=-1\end{array}\right.\)
- step3: Simplify:
\(\left\{ \begin{array}{l}12-10x_{2}-5x_{3}=2\\20-7x_{2}=-1\end{array}\right.\)
- step4: Solve the equation:
\(\left\{ \begin{array}{l}12-10x_{2}-5x_{3}=2\\x_{2}=3\end{array}\right.\)
- step5: Substitute the value of \(x_{2}:\)
\(12-10\times 3-5x_{3}=2\)
- step6: Simplify:
\(-18-5x_{3}=2\)
- step7: Move the constant to the right side:
\(-5x_{3}=2+18\)
- step8: Add the numbers:
\(-5x_{3}=20\)
- step9: Change the signs:
\(5x_{3}=-20\)
- step10: Divide both sides:
\(\frac{5x_{3}}{5}=\frac{-20}{5}\)
- step11: Divide the numbers:
\(x_{3}=-4\)
- step12: Substitute the values of \(x_{2},x_{3}:\)
\(x_{1}=4-2\times 3-\left(-4\right)\)
- step13: Simplify:
\(x_{1}=2\)
- step14: Calculate:
\(\left\{ \begin{array}{l}x_{1}=2\\x_{2}=3\\x_{3}=-4\end{array}\right.\)
- step15: Check the solution:
\(\left\{ \begin{array}{l}x_{1}=2\\x_{2}=3\\x_{3}=-4\end{array}\right.\)
- step16: Rewrite:
\(\left(x_{1},x_{2},x_{3}\right) = \left(2,3,-4\right)\)
The solution to the set of equations using the matrix method is \( (x_{1}, x_{2}, x_{3}) = (2, 3, -4) \).
Quick Answer
The solution is \( (x_{1}, x_{2}, x_{3}) = (2, 3, -4) \).
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