Given the system of linear equations ax + by = c and ax + by = 0, determine the number of solutions based on the value of c.

Both equations have the same coefficients for $x$ ($a$) and $y$ ($b$). This means they represent lines with the same slope.

The constant term in equation (1) is $c$, and in equation (2) it is 0. The relationship between the two lines (and thus the number of solutions) depends on the value of $c$.

• Case 1: If $c \neq 0$ If $c$ is any non-zero number, then $ax + by$ cannot simultaneously equal $c$ and 0. This means the two equations represent two distinct parallel lines. Parallel lines never intersect, so there is no solution to the system.

• Case 2: If $c = 0$ If $c$ is equal to 0, then both equations become $ax + by = 0$. This means the two equations are identical. They represent the same line. In this case, there are infinitely many solutions, as any point $(x, y)$ that satisfies $ax + by = 0$ is a solution to the system.

The number of solutions depends on the value of $c$: