If the quantity given by 3m−1+a/ 5m+6 is equal to one and the same number (constant) for every real m, then what is the value of a?

(3m-1+a) / (5m+6) = k (constant)

3m-1+a = 5km + 6k

a = 5km-3m + 6k + 1

a = m(5k - 3) + 6k+1

The quantity, a, can have a unique value if and only if 5k-3 = 0 or k = 3/5. Therefore:

a = m(0) + 6(3/5) + 1 = 23/5.

Ans: a = 23/5 = 4.6

======================

Check:

(3m - 1 + 23/5) / (5m + 6)

 = (3m + 18/5) / (5m+6)

 = (15m + 18)/[5(5m+6)]

 = 3(5m+6)/[5(5m+6)]

 = 3/5

Parentheses are important when typing an expression on a single line. You apparently meant to write:

(3m - 1 + a) / (5m + 6)

If that is equal to a constant, then the numerator is some constant multiplier of the denominator.

Let's multiply that by 5/3 and it won't affect the result being a constant. We'll also make it so that we have 15m in both the numerator and denominator. Since m can be anything, the way this will be a constant is if the numerator and denominator are equal.

15m - 5 + 5a = 15m + 18

We can cancel 15m and then equate the rest.

5a - 5 = 18

5a = 23

a = 23/5

a = 4 3/5 (or 4.6)

P.S. The exception is if the value of m = -6/5 because then the denominator will be 0. So this is really only true for all real values of m *other than* -6/5.

3m - 1 + (a/5)m + 6

= (3 + 0.2a)m + 5

If that's a constant, then (3 + 0.2a)m is a constant. The only way that's possible is if:

3 + 0.2a = 0

0.2a = -3

a = -3/0.2

a = -15