Requirements for $b$ and $c$ in a equation system

Question

I have following system of equations.

\begin{align} cx + 2y &=1 \\ \frac23 x -y &=b \end{align} If $b$ and $c$ are real numbers , what are the requirements for $b$ and $c$ to get only one solution at the end?

Answer 1

From the first equation $y = (1-cx)/2$. If you substitute that in the second, \begin{align} 2/3x-(1-cx)/2 &= b \\ 2/3x - 1/2 + (c/2)x &= b \\ x(2/3+c/2) &= b + 1/2 \end{align}

From this, you can see that if $b+1/2 = 2/3 + c/2 = 0$, $x$ can take any value, thus not a unique solution. If you have that $b+1/2 \ne 0$ and $2/3+c/2 = 0$, then no value of $x$ will satisfy the equation.