Solve the system of equations. (a) Find the values of c for...

Solve the system of equations.
(a) Find the values of $c$ for which this system has an infinite number of solutions.
$$\begin{aligned}2 x-4 y &=6 \\-3 x+6 y &=c\end{aligned}$$
(b) Find the values of $c$ for which the system in part (a) has no solutions.

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Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. The solutions to the system are the values of the variables that satisfy all equations simultaneously. In the geometric interpretation, each equation represents a line in the plane, and the solution corresponds to the intersection point(s) of these lines.

Coefficient Ratios in Determining Dependency

The ratios of the coefficients in a system of linear equations indicate the relationship between the lines. When the ratios of the coefficients of corresponding variables in the equations are equal, the lines are either coincident (infinitely many solutions) or parallel (no solution), depending on whether the constant terms are also in the same ratio. This comparison is used to determine if the system is dependent or inconsistent.

Consistent and Inconsistent Systems

A consistent system has at least one solution, either exactly one or infinitely many if the equations represent the same geometric line. In contrast, an inconsistent system has no solutions, typically when the equations represent parallel lines that never intersect. Checking for consistency involves comparing both the coefficients and the constants in the equations.

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