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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

Suppose an experiment leads to the following system of equations:

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{249}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.843\end{aligned}\) (3)

  1. Solve system (3), and then solve system (4), below, in which the data on the right have been rounded to two decimal places. In each case, find the exactsolution.

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{25}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.8{\bf{4}}\end{aligned}\) (4)

  1. The entries in (4) differ from those in (3) by less than .05%. Find the percentage error when using the solution of (4) as an approximation for the solution of (3).
  1. Use your matrix program to produce the condition number of the coefficient matrix in (3).

Determine which of the matrices in Exercises 7鈥�12areorthogonal. If orthogonal, find the inverse.

11. \(\left( {\begin{aligned}{{}}{2/3}&{2/3}&{1/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\)

Explain why a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\) in \({\mathbb{R}^5}\) must be linearly independent when \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\) is linearly independent and \({{\mathop{\rm v}\nolimits} _4}\) is not in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

Consider the problem of determining whether the following system of equations is consistent for all \({b_1},{b_2},{b_3}\):

\(\begin{aligned}{c}{\bf{2}}{x_1} - {\bf{4}}{x_2} - {\bf{2}}{x_3} = {b_1}\\ - {\bf{5}}{x_1} + {x_2} + {x_3} = {b_2}\\{\bf{7}}{x_1} - {\bf{5}}{x_2} - {\bf{3}}{x_3} = {b_3}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of Span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\). Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a plane in \({\mathbb{R}^3}\).
See all solutions

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