To find the number of integral values of m so that the abscissa (x-coordinate) of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is an integer, we will solve the system of linear equations and find the conditions under which the solution is an integer.
The first equation is: 3x + 4y = 9
The second equation is: y = mx + 1
Substituting the value of y from the second equation into the first equation:
3x + 4(mx + 1) = 9
Simplifying this: 3x + 4mx + 4 = 9
(3 + 4m)x = 9 - 4
(3 + 4m)x = 5
Thus, the solution for x is: x = \frac{5}{3 + 4m}
For x to be an integer, 3 + 4m must be a divisor of 5. The divisors of 5 are 1, -1, 5, and -5.
Let's evaluate each possibility:
If 3 + 4m = 1, then 4m = 1 - 3 = -2 \Rightarrow m = -\frac{1}{2} (Not an integer).
If 3 + 4m = -1, then 4m = -1 - 3 = -4 \Rightarrow m = -1 (This is an integer).
If 3 + 4m = 5, then 4m = 5 - 3 = 2 \Rightarrow m = \frac{1}{2} (Not an integer).
If 3 + 4m = -5, then 4m = -5 - 3 = -8 \Rightarrow m = -2 (This is an integer).
Therefore, the integral values of m that satisfy the condition are m = -1 and m = -2.
