函数$y=ax+b$和$y=kx$的图象交于点$P\left(-2,-1\right)$，

即$x=-2$，$y=-1$同时满足两个一次函数的解析式.

所以关于$x$，$y$的方程组$\left\{\begin{array}{l}y=ax+b\\y=kx\end{array}\right.$的解是$\left\{\begin{array}{l}x=-2\\y=-1\end{array}\right.$.

故答案为：$\left\{\begin{array}{l}x=-2\\y=-1\end{array}\right.$.

函数$y=ax+b$和$y=kx$的图象交于点$P\left(-2,-1\right)$，

即$x=-2$，$y=-1$同时满足两个一次函数的解析式.

所以关于$x$，$y$的方程组$\left\{\begin{array}{l}y=ax+b\\y=kx\end{array}\right.$的解是$\left\{\begin{array}{l}x=-2\\y=-1\end{array}\right.$.

故答案为：$\left\{\begin{array}{l}x=-2\\y=-1\end{array}\right.$.