Consider the following system of equations.
- 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 x_1+kx_2=1}
- 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 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:
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| 1. To solve a system of equations, we turn the system into an augmented matrix and
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- row reduce that matrix to determine the solution.
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| 2. For a system to have a unique solution, we need to have no free variables.
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Solution:
| Step 1:
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| To begin with, we turn this system into an augmented matrix.
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| Hence, we get
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- 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].}
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| Now, when we row reduce this matrix, we get
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- 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].}
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| Step 2:
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| To guarantee a unique solution, our matrix must contain two pivots.
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| So, we must have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle -3k+5\neq 0.}
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| Hence, we must have
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- 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}.}
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| 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}.}
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| Final Answer:
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| 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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