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Use elementary operations to find the general solution of each of the following systems
Chapter 1, Problem 1(choose chapter or problem)
Use elementary operations to find the general solution of each of the following systems of equations. Use the method of Example 1 as a prototype.
a.
\(\begin{aligned}
x_{1}+x_{2} & =1 \\
x_{1}+2 x_{2}+x_{3} & =1 \\
x_{2}+2 x_{3} & =1
\end{aligned}\)
b.
\(\begin{array}{r}
x_{1}+2 x_{2}+3 x_{3}=1 \\
2 x_{1}+4 x_{2}+5 x_{3}=1 \\
x_{1}+2 x_{2}+2 x_{3}=0
\end{array}\)
c.
\(\begin{aligned}
3 x_{1}-6 x_{2}-x_{3}+x_{4} & =6 \\
-x_{1}+2 x_{2}+2 x_{3}+3 x_{4} & =3 \\
4 x_{1}-8 x_{2}-3 x_{3}-2 x_{4} & =3
\end{aligned}\)
Questions & Answers
(3 Reviews)
QUESTION:
Use elementary operations to find the general solution of each of the following systems of equations. Use the method of Example 1 as a prototype.
a.
\(\begin{aligned}
x_{1}+x_{2} & =1 \\
x_{1}+2 x_{2}+x_{3} & =1 \\
x_{2}+2 x_{3} & =1
\end{aligned}\)
b.
\(\begin{array}{r}
x_{1}+2 x_{2}+3 x_{3}=1 \\
2 x_{1}+4 x_{2}+5 x_{3}=1 \\
x_{1}+2 x_{2}+2 x_{3}=0
\end{array}\)
c.
\(\begin{aligned}
3 x_{1}-6 x_{2}-x_{3}+x_{4} & =6 \\
-x_{1}+2 x_{2}+2 x_{3}+3 x_{4} & =3 \\
4 x_{1}-8 x_{2}-3 x_{3}-2 x_{4} & =3
\end{aligned}\)
Step 1 of 3
a)
\(\begin{array}{l}x_1+x_2=1\ ....\ \left(1\right)\\ x_1=x_2+x_3=1\ ....\ \left(2\right)\\ x_2+2x-3=1\ ....\ \left(3\right)\end{array}\)
Subtracting equation (1) from equation (2) the equation reduces to
\(\begin{array}{l} x_{1}+x_{2}=1 \ldots(1) \\ x_{2}+x_{3}=0 \ldots(2) \\ x_{2}+2 x_{3}=1 \ldots(3) \end{array}\)
Again subtracting equation (2) from equation (3) we get
\(\begin{array}{l} x_{1}+x_{2}=1 \ldots(1) \\ x_{2}+x_{3}=0 \ldots(2) \\ x_{3}=1 \ldots \ldots \ldots(3) \end{array}\)
Substitute (3) in (2) we get
\(\begin{array}{l} x_{2}+1=0 \\ x_{2}=-1 \end{array}\)
Again from (1) we get
\(\begin{array}{l} x_{1}-1=1 \\ x_{1}=2 \end{array}\)
Thus the general solution is,
\(x=\left(x_{1}, x_{2}, x_{3}\right)=(2,-1,1)\)
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Review this written solution for 964097) viewed: 701 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 1.4 - Problem 1
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Review this written solution for 964097) viewed: 701 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 1.4 - Problem 1
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Review this written solution for 964097) viewed: 701 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 1.4 - Problem 1
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