031 Review Part 2, Problem 11

Consider the following system of equations.

$x_{1}+kx_{2}=1$

$3x_{1}+5x_{2}=2k$

Find all real values of  $k$  such that the system has only one solution.

Foundations:
1. To solve a system of equations, we turn the system into an augmented matrix and
row reduce that matrix to determine the solution.
2. For a system to have a unique solution, we need to have no free variables.

Solution:

Step 1:
To begin with, we turn this system into an augmented matrix.
Hence, we get
$\left[{\begin{array}{cc|c}1&k&1\\3&5&2k\end{array}}\right].$

Step 2:

Final Answer:

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