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  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{cc|c} 1 & k & 1 \\ 3 & 5 & 2k \end{array}\right].}
Now, when we row reduce this matrix, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\begin{array}{cc|c} 1 & k & 1 \\ 3 & 5 & 2k \end{array}\right] \sim \left[\begin{array}{cc|c} 1 & k & 1 \\ 0 & -3k+5 & -3+2k \end{array}\right].}

Step 2:
To guarantee a unique solution, our matrix must contain two pivots.
So, we must have  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3k+5\ne 0.}
Hence, we must have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\ne \frac {5}{3}.}
Therefore,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}   can be any real number except  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5}{3}.}

Final Answer:

The system has only one solution when  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}   is any real number except  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5}{3}.}

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