Find values of K for which k has no solution, many solutions a unique solution

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The discussion focuses on determining the values of K for which the system of equations x + ky = 1 and kx + y = 1 has no solution, many solutions, or a unique solution. Two methods are suggested: solving the system for x and y as functions of k to identify inconsistent values, and using a graphical approach to analyze the relationships between the lines represented by the equations. The conditions for solutions are clarified: if the lines intersect at a point, there is a unique solution; if they overlap, there are many solutions; and if they are parallel, there are no solutions. The problem statement is refined to emphasize the search for K values affecting the solution types. Understanding these concepts is essential for solving the system effectively.
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Homework Statement



Find values of K for which k has no solution, many solutions a unique solution

Homework Equations



x + ky = 1
kx + y =1


The Attempt at a Solution



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What in the world is an expression like

\frac{\binom{x+ky=1}{kx+y=1}}{y}

supposed to mean?

Do you know Cramer's rule? Can you find a k that makes the equations dependent? Inconsistent?
 
Method I: Try to solve the system for x and y as a function of k. Now check for which values of k the solution makes no sense (for example, if you have to divide by 0). Those are the values for which the system is inconsistent, so no solutions.

Method II: graphical method. The two equations give two lines in the plane. If the cut at a point, the system is fine. If they're the same line, the system has many solutions. If they're parallel, the system has no solutions.
 
The problem statement is shown as
judahs_lion said:
Find values of K for which k has no solution, many solutions a unique solution
It makes more sense as "Find values of k for which the system of equations has no solution, many solutions, a unique solution."
 

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