Find the values of a and b so that the solution of the linear system is (−9, 1) ax+by=−31 ax−by=−41 show steps.

Since the point (-9,1) is a solution to the given system of equations, therefore, this point satisfies both equations.

For the first equation:
ax + by = -31
a(-9) + b(1) = -31
-9a +b = -31
This can be rewritten as:
b = -31 + 9a > equation I

For the second equation:
ax - by = -41
a(-9) - b(1) = -41
-9a - b = -41 > equation II

Substitute with equation I in equation II to get a as follows:
-9a - b = -41
-9a - (-31 + 9a ) = -41
-9a + 31 - 9a = -41
-18a = -41-31
-18a = -72
a = 4

Substitute with the value of a in equation I to get b as follows:
b = -31 + 9a
b = -31 + 9(4)
b = -31 + 36
b = 5

Based on the above calculations:
a = 4
b = 5