For the linear function $f(x)=m x+b, f(-7)=19$, and $f(4)=-3$. Find $m$ and $b$. \[ \begin{array}{l} m=\square \\ b=\square \end{array} \]

Step 1 ：We are given a linear function $f(x)=m x+b$, where $f(-7)=19$ and $f(4)=-3$. We are asked to find the values of $m$ and $b$.

Step 2 ：We can set up two equations based on the given information: $-7m + b = 19$ and $4m + b = -3$.

Step 3 ：Solving this system of equations gives us the values of $m$ and $b$.

Step 4 ：The solution to the system of equations is $m = -2$ and $b = 5$. This means that the slope of the line is -2 and the y-intercept is 5. This is consistent with the given information in the problem.

Step 5 ：Final Answer: \[ \begin{array}{l} m=\boxed{-2} \ b=\boxed{5} \end{array} \]