In Exercises 19 and 20, choose h and k such that the system...

In Exercises 19 and $20,$ choose $h$ and $k$ such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part.
$\begin{aligned} x_{1}+3 x_{2} &=2 \\ 3 x_{1}+h x_{2} &=k \end{aligned}$

Key Concepts

Parameterization in Linear Systems

Parameterization involves including unknown parameters (like h and k) in the coefficients or constant terms of the equations. This allows us to explore different cases—such as when a system has no solution, a unique solution, or infinitely many solutions—by varying the parameters. It is a powerful way to understand how changes in coefficients affect the overall solution of the system.

Infinitely Many Solutions

Infinitely many solutions occur when the system is dependent, meaning the equations represent the same line. In this scenario, every point on the line is a solution, which indicates that the second equation is just a multiple of the first.

Consistency of a System

A system is consistent if there is at least one solution and inconsistent if there is no solution. In the context of linear equations, inconsistency often occurs when the lines are parallel (i.e., they have the same slope) but do not coincide, leading to a situation where the equations contradict each other.

System of Linear Equations

A system of linear equations is a set of equations involving the same variables. The solution is any common set of variable values that satisfies all equations simultaneously. Analyzing such systems helps determine whether the lines represented by the equations intersect, overlap, or are parallel.

Unique Solution

A unique solution occurs when the system of equations intersects at exactly one point. This happens when the equations are independent, meaning that the lines represented by the equations have different slopes and thus intersect at only a single point.

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