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# Find the general solution of each of the following equations (presented, as in the text

**Chapter 1, Problem 3**

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**QUESTION:**

Find the general solution of each of the following equations (presented, as in the text, as a combination of an appropriate number of vectors)

a. \(x_{1}-2 x_{2}+3 x_{3}=4\left(\text { in } \mathbb{R}^{3}\right)\)

b. \(x_{1}+x_{2}-x_{3}+2 x_{4}=0\left(\text { in } \mathbb{R}^{4}\right)\)

c. \(x_{1}+x_{2}-x_{3}+2 x_{4}=5\left(\text { in } \mathbb{R}^{4}\right)\)

d. \(x_{1}-2 x_{2}+3 x_{3}=4\left(\text { in } \mathbb{R}^{4}\right)\)

e. \(x_{2}+x_{3}-3 x_{4}=2\left(\text { in } \mathbb{R}^{4}\right)\)

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(3 Reviews)

**QUESTION:**

Find the general solution of each of the following equations (presented, as in the text, as a combination of an appropriate number of vectors)

a. \(x_{1}-2 x_{2}+3 x_{3}=4\left(\text { in } \mathbb{R}^{3}\right)\)

b. \(x_{1}+x_{2}-x_{3}+2 x_{4}=0\left(\text { in } \mathbb{R}^{4}\right)\)

c. \(x_{1}+x_{2}-x_{3}+2 x_{4}=5\left(\text { in } \mathbb{R}^{4}\right)\)

d. \(x_{1}-2 x_{2}+3 x_{3}=4\left(\text { in } \mathbb{R}^{4}\right)\)

e. \(x_{2}+x_{3}-3 x_{4}=2\left(\text { in } \mathbb{R}^{4}\right)\)

**ANSWER:**

Step 1 of 3

a)Because there are variables in this one equation,specifying two determines the third.So let’s set two of them equal to arbitrary variable names.

Let \(x_{2}=s\)

and \(x_{3}=t\)

Then we see that \(x_{1}-x_{2}+3 x_{3}=4=x_{1}=4+2 s-3 t\).

Thus the solution set is all vectors of the form

\(X=\left(x_{1}, x_{2}, X-3\right)=(4+2 s-3 t, s, t)=(4,0,0)+s(2,1,0)+t(-3,0,1)\)

b)

In the rest of these we proceed exactly in (a).

So Let \(x_{2}=r\)

\(x_{3}=s\)

and \(x_{4}=t\).

Then

\(x=(-r+s-2 t, r, s, t)=r(-1,1,0,0)+s(1,0,1,0)+t(-2,0,0,1)\)

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### Review this written solution for 964024) viewed: 285 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 1.3 - Problem 3

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### Review this written solution for 964024) viewed: 285 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 1.3 - Problem 3

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### Review this written solution for 964024) viewed: 285 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 1.3 - Problem 3

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