Chapter 1 - Linear Equations in Linear Algebra - Supplementary Exercises - Page 89: 2

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\[ a=0, b=0 \] If $a=0$ and $b=0,$ then this equation will be true for all values of $x .$ This is since $0 x=0$ for all $x$ \[ a=0, b \neq 0 \] If $a=0$ and $b \neq 0$ then the equation will have no solutions. Since $a=0, a x=0$ for all $x,$ but $b \neq 0,$ therefore, $a x \neq b$ for all $x$ \[ a \neq 0 \] If $a \neq 0,$ then this equation will have a unique solution no matter what the value of $b$ is. That unique solution will be $x=\frac{b}{a} .$ since $a \neq 0,$ we can divide by $a,$ which is what allows for this solution. \[ a \cdot \frac{b}{a}=b \]