9.1. For what values of the number $a$ do the three graphs $y=a x+a, y=x$ and $y=2-2 a x$ intersect at one point
Answer: when $a=1 / 2$ and $a=-2$. Solution. The problem reduces to solving the given three equations with three unknowns $x, y$ and $a$. Substitute the expression for $y$ from the second equation $y=x$ into the first and third equations: $\left\{\begin{array}{l}x=a x+a \\ x=2-2 a x\end{array}\right.$. Express from the first equation of the system $a=\frac{x}{x+1}$ and substitute into the second. As a result, we get a quadratic equation $x(x+1)=$ $2(x+1)-2 x^{2} \Leftrightarrow 3 x^{2}-x-2=0$. Hence, $x_{1,2}=(1 \pm 5) / 6$. We have two solutions $x_{1}=1$ and $x_{2}=-$ $2 / 3$. The corresponding values of $a$ are $a_{1}=1 / 2$ and $a_{2}=-2$. (Remark:; no need to check if you notice that when solving we multiplied the equation by ( $x+1$ ), but there is no root $x=-1)$