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Chapter 1: Q30E (page 1)

Give an example of an inconsistent underdetermined system of two equations in three unknowns.

Short Answer

Expert verified

One of the examples of an inconsistent underdetermined system of two equations in three unknowns is

Step by step solution

01

Inconsistent underdetermined system

The underdetermined system is a system of equations with fewer equations than unknowns. When this system becomes inconsistent, it is known as an inconsistent underdetermined system.

02

Consider the example

Consider the system of two equations in three unknowns, that is, an underdetermined system:

\(\begin{array}{c}x + 2y + z = 2\\ - x - 2y - z = 1\end{array}\)

03

Apply the addition operator

Add the above two equations to get:

\(0 = 3\)

This is not possible. So, our system is inconsistent.

04

Conclusion

Hence, the system given below is an example of an inconsistent underdetermined system of two equations in three unknowns.

\(\begin{array}{c}x + 2y + z = 2\\ - x - 2y - z = 1\end{array}\)

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Most popular questions from this chapter

In Exercises 21 and 22, find a parametric equation of the line \(M\) through \({\mathop{\rm p}\nolimits} \) and \({\mathop{\rm q}\nolimits} \). [Hint: \(M\) is parallel to the vector \({\mathop{\rm q}\nolimits} - p\). See the figure below.]

22. \({\mathop{\rm p}\nolimits} = \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right]\), \(q = \left[ {\begin{array}{*{20}{c}}0\\{ - 4}\end{array}} \right]\)

In Exercises 5鈥�8, determine if the columns of the matrix form a

linearly independent set. Justify each answer.

7. \(\left[ {\begin{array}{*{20}{c}}1&4&{ - 3}&0\\{ - 2}&{ - 7}&5&1\\{ - 4}&{ - 5}&7&5\end{array}} \right]\)

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot position in each column. Explain why the system has a unique solution.

A steam plant burns two types of coal: anthracite (A) and bituminous (B). For each ton of A burned, the plant produces 27.6 million of Btu of heat, 3100 grams (g) of sulphur dioxide, and 250g of particulate matter (solid-particle pollutants). For each ton of B burned, the plant produces 30.2 million Btu, 6400g of sulphur dioxide, and 360g of particulate matter.

  1. How much heat does the steam plant produce when it burns \({x_1}\) tons of \(A\) and \({x_2}\) tons of \(B\).
  2. Suppose a vector that lists the amounts of heat, sulphur dioxide, and particulate matter describes the output of the steam plant. Express this output as a linear combination of two vectors, assuming that the plant burns \({x_1}\) tons of \(A\) and \({x_2}\) tons of \(B\).
  3. [M] Over a certain time period, the steam plant produced 162 million Btu of heat, 23,610 g of sulphur dioxide, and 1623 g of particulate matter. Determine how many tons of each type of coal the steam plant must have burned. Include a vector equation as part of your solution.

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

16. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\)
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