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Chapter 1: Q2Q (page 1)

Let a and b represent real numbers. Describe the possible solution sets of the (linear) equation \(ax = b\). (Hint:The number of solutions depends upon a and b.)

Short Answer

Expert verified

The solution is unique when \(a \ne 0\). The solution is not possible when \(a = 0\) and \(b \ne 0\). The number of solutions is infinite when \(a = 0\) and \(b = 0\).

Step by step solution

01

Re-arrange the linear equation

Consider the linear equation\(ax = b\), where a and b are real numbers.

Re-arrange this equation (divide both sides by a) to obtain the value of x in terms of a and b, as shown below:

\(\begin{aligned}{c}\frac{{ax}}{a} = \frac{b}{a}\\x = \frac{b}{a}\end{aligned}\)

02

Describe the possible solution sets

Consider the equation\(x = \frac{b}{a}\). When\(a \ne 0\), the solution is\(x = \frac{b}{a}\).

Here, \(\frac{b}{a}\) is the unique solution for the linear equation when \(a \ne 0\).

03

Describe the possible solution sets

Consider the case when\(a = 0\)and\(b \ne 0\).

The solution is not possible because the denominator of the equation\(x = \frac{b}{a}\)cannot be 0, that is, \(\left( 0 \right)x = 0 \ne b\).

Thus, the solution is not possible.

04

Describe the possible solution sets

Consider the case when\(a = 0\)and\(b = 0\).

There are infinitely many solutions for the equation\(ax = b\)because\(\left( 0 \right)x = 0 = b\).

Thus, the number of solutions is infinite.

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Most popular questions from this chapter

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes鈥攖o the left, above, to the right, and below. For instance,

\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)

33. Write a system of four equations whose solution gives estimates

for the temperatures \({T_1},...,{T_4}\).

Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.

6. \(\left( {\begin{aligned}{*{20}{c}}1&{ - 6}&4&0&{ - 1}\\0&2&{ - 7}&0&4\\0&0&1&2&{ - 3}\\0&0&3&1&6\end{aligned}} \right)\)

Let \(A = \left[ {\begin{array}{*{20}{c}}2&0&6\\{ - 1}&8&5\\1&{ - 2}&1\end{array}} \right]\), let \(b = \left[ {\begin{array}{*{20}{c}}{10}\\3\\3\end{array}} \right]\) , and let \(W\) be the set of all linear combinations of the columns of \(A\).

  1. Is \(b\) in \(W\)?
  2. Show that the third column of \(A\) is in \(W\).

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

Suppose \(a,b,c,\) and \(d\) are constants such that \(a\) is not zero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a,b,c,\) and \(d\)? Justify your answer.

28. \(\begin{array}{l}a{x_1} + b{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)
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