Question

For certain values of k and m, the system

3a + 2b = 2 - 7a + 5b

6a + 2b = k - 5a + mb + 17 - 11a + 18b

has infinitely many solutions (a,b). What are k and m?

Answer 1

To find k and m, we need to find the conditions for which the system has infinitely many solutions. This occurs when the two equations are dependent, meaning that one equation is a multiple of the other equation.

To determine if the equations are dependent, we can multiply the first equation by a constant to see if it matches the second equation.

If we multiply the first equation by 3, we get:

9a + 6b = 6 - 21a + 15b

Comparing this equation to the second equation, we see that the coefficients of a and b match up. However, the constant terms do not match. To make the constant terms match, we must have:

6 - 21a + 15b = k - 5a + mb + 17 - 11a + 18b

Simplifying this equation, we get:

6 - 21a + 15b = k - 5a + mb + 17 - 11a + 18b
-14a + 3b = k + mb + 11

So, for the equations to be dependent, we need:

-14a + 3b = k + mb + 11

Note that the left side of this equation does not depend on k or m. Therefore, k and m can take on any values and the system will still have infinitely many solutions.

Therefore, there are no specific values of k and m that satisfy the given conditions.