Chapter 7.2, Problem 91E

(a)

To determine

To find: the value of $k$ for which the given system of linear equations has infinitely many solutions.

Answer to Problem 91E

The value of $k$ for which the given system of linear equations has infinitely many solutions is $18$ .

Explanation of Solution

Given information:

A system of linear equations:

  $x+3y=9$  ...(1)

  $2x+6y=k$  ...(2)

Calculation:

Compare equation (1) with $a_{1}x+b_{1}y=c_1$ to get the value of $a_1,b_1{\text{ and c}}_1$ .

  $\begin{array}{l}{a}_1=1\\ b_1=3\\ c_1=9\end{array}$

Compare equation (2) with $a_{2}x+b_{2}y=c_2$ to get the value of $a_2,b_2{\text{ and c}}_2$ .

  $\begin{array}{l}{a}_2=2\\ b_2=6\\ c_2=k\end{array}$

For infinitely many solutions:

  $\begin{array}{l}\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\\ \frac{1}{2}=\frac{3}{6}=\frac{9}{k}\\ \frac{1}{2}=\frac{9}{k}\text{ and }\frac{3}{6}=\frac{9}{k}\\ \frac{1}{2}=\frac{9}{k}\end{array}$

  $\begin{array}{l}k=2\times 9\\ k=18\end{array}$

Therefore, the value of $k$ for which the given system of linear equations has infinitely many solutions is $18$ .

(b)

To determine

To find: the value of $k$ for which the given system of linear equations has no solution.

Answer to Problem 91E

The value of $k$ must not be equal to $18$ for the given system of linear equations has no solution.

Explanation of Solution

Given information:

A system of linear equations:

  $x+3y=9$  ...(1)

  $2x+6y=k$  ...(2)

Calculation:

Compare equation (1) with $a_{1}x+b_{1}y=c_1$ to get the value of $a_1,b_1{\text{ and c}}_1$ .

  $\begin{array}{l}{a}_1=1\\ b_1=3\\ c_1=9\end{array}$

Compare equation (2) with $a_{2}x+b_{2}y=c_2$ to get the value of $a_2,b_2{\text{ and c}}_2$ .

  $\begin{array}{l}{a}_2=2\\ b_2=6\\ c_2=k\end{array}$

For no solution:

  $\begin{array}{l}\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}\\ \frac{1}{2}=\frac{3}{6}\ne \frac{9}{k}\\ \frac{1}{2}\ne \frac{9}{k}\text{ and }\frac{3}{6}\ne \frac{9}{k}\\ \frac{1}{2}\ne \frac{9}{k}\end{array}$

  $\begin{array}{l}k\ne 2\times 9\\ k\ne 18\end{array}$

Therefore, the value of $k$ must not be equal to $18$ for the given system of linear equations has no solution.