Solve the Problem , And $\mathbf { b } = \left[ \begin{array} { r } - 1 \\ 1 \\ 2 \end{array} \right]$

question 40

Multiple Choice

Solve the problem.
-Let $\mathbf { a } _ { \mathbf { 1 } } = \left[ \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right] , \mathbf { a } _ { \mathbf { 2 } } = \left[ \begin{array} { r } - 3 \\ - 4 \\ 1 \end{array} \right] , \mathbf { a } _ { 3 } = \left[ \begin{array} { l } 2 \\ 1 \\ 6 \end{array} \right]$ , and $\mathbf { b } = \left[ \begin{array} { r } - 1 \\ 1 \\ 2 \end{array} \right]$ .
Determine whether $\mathbf { b }$ can be written as a linear combination of $\mathbf { a } _ { \mathbf { 1 } } , \mathbf { a } _ { \mathbf { 2 } }$ , and $\mathbf { a } _ { 3 }$ . In other words, determine whether weights $x _ { 1 } , x _ { 2 }$ , and $x _ { 3 }$ exist, such that $x _ { 1 } a _ { 1 } + x _ { 2 } a _ { 2 } + x _ { 3 } a _ { 3 } = b$ . Determine the weights $x _ { 1 } , x _ { 2 }$ , and $x _ { 3 }$ if possible.

• A) \(x _ { 1 } = 2 , x _ { 2 } = 1 , x _ { 3 } = 0\)
• B) \(x _ { 1 } = - 3 , x _ { 2 } = 0 , x _ { 3 } = 1\)
• C) No solution
• D) \(x _ { 1 } = - 2 , x _ { 2 } = - 1 , x _ { 3 } = 1\)

---

Definitions:

Union Membership

Involvement in an organized association of workers formed to protect and further their rights and interests in the workplace.

Labor Force

The total number of people, both employed and unemployed, who are actively looking for work in the economy.

Inclusive Union

A labor union that aims to represent all workers within a particular industry or workplace, regardless of job or skill level, focusing on broad inclusivity.

Labor Supply Curve

A visual chart that illustrates the connection between the rate of pay and the amount of labor that workers are prepared to offer.