Determine whether the statement below is true or false. Justify the answer.

The solution set of the linear system whose augmented matrix is $\left[\begin{array}{cccc}{a}_1& a_2& a_3& b\end{array}\right]$ is the same as the solution set of the equation $x_1{a}_1+x_2{a}_2+x_3{a}_3=b.$

Choose the correct answer below.

A$.$ False. The solution set for an augmented matrix is not the same as the solution set of an equation.

B$.$ True. The augmented matrix for $x_1{a}_1+x_2{a}_2+x_3{a}_3=b$ is $\left[\begin{array}{cccc}{x}_1& x_2& x_3& b\end{array}\right].$

C$.$ False. The augmented matrix for $x_1{a}_1+x_2{a}_2+x_3{a}_3=b$ is not $\left[\begin{array}{cccc}{a}_1& a_2& a_3& b\end{array}\right].$

D$.$ True. The augmented matrix for $x_1{a}_1+x_2{a}_2+x_3{a}_3=b$ is $\left[\begin{array}{cccc}{a}_1& a_2& a_3& b\end{array}\right].$