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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

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

34. \(T\left( {{x_1},{x_2}} \right) = \left( {6{x_1} - 8{x_2}, - 5{x_1} + 7{x_2}} \right)\)

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

Let \(u = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\) and \(v = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\). Show that \(\left[ {\begin{array}{*{20}{c}}h\\k\end{array}} \right]\) is in Span \(\left\{ {u,v} \right\}\) for all \(h\) and\(k\).

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation, and suppose \(T\left( u \right) = {\mathop{\rm v}\nolimits} \). Show that \(T\left( { - u} \right) = - {\mathop{\rm v}\nolimits} \).

Let \(A\) be a \(3 \times 3\) matrix with the property that the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^3}\) into \({\mathbb{R}^3}\). Explain why transformation must be one-to-one.
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