Question

Suppose you have the following system of equations, where a and b are real number constants:

3x - 2y = -1
ax + by = -2

What values of a and b, if any, would yield a system with no solutions?
(a) a = 3, b = -2
(b) a = 3, b = 2
(c) a = -3, b = 2
(d) a = -3, b = -2

Answer 1

For the system of equations to have no solutions, the lines must be parallel, meaning the ratio of the coefficients is the same but with different constant terms. The correct choice is (a) a = 3, b = -2.

Step-by-step explanation:

A system of equations will have no solutions if the two equations represent parallel lines. This happens when the ratio of the coefficients of the x and y terms of the equations is the same, but the constants on the right side of the equation are different.

The first equation is 3x - 2y = -1. In order for the second equation to be parallel to the first, it must have the form ax + by = c, where a/3 = b/(-2), but with a different constant term on the right side. For this system, the values that make the lines parallel (and thus, yield no solutions) are a = 3 and b = -2, following the condition a/3 = b/(-2) which simplifies to a = -3b.

The correct option is therefore (a) a = 3, b = -2.