If the simultaneous linear equations in x and g, y = kx + m and y= (2k - 1)x +4 have at least one solution, where k and m are constants, find the possible values of k and m

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Identical Lines Case: For the two linear equations y = kx + m and y = (2k - 1)x + 4 to have at least one solution, they must either be the same line (infinite solutions) or intersect at a single point (one solution). To find the possible values of k and m, we can equate the two equations since they both equal y.Identical Lines Case: Setting the equations equal to each other gives us kx + m = (2k - 1)x + 4.Intersecting Lines Case: To find the possible values of k and m, we need to consider two cases: either the lines are identical (infinite solutions) or they intersect at exactly one point (one solution). If the lines are identical, the coefficients of x and the constant terms must be equal in both equations.Identical Lines Solution: For the lines to be identical, we must have k = 2k - 1 and m = 4. Solving the first equation for k gives us k = 1. Plugging k = 1 into the second equation, we get m = 4.Intersecting Lines Solution: If the lines are not identical but intersect at one point, then the slopes (coefficients of x) must be different. This means k ≠ 2k - 1. Solving for k, we get k ≠ 1. In this case, m can be any real number since the lines will intersect at one point regardless of the value of m.Final Conclusion: Therefore, the possible values for k and m are k = 1 and m = 4 for the case of identical lines, and for the case of intersecting lines, k can be any real number except 1, and m can be any real number.

If the simultaneous linear equations in $x$ and $g$ , $y = kx + m$ and $y= (2k - 1)x +4$ have at least one solution, where $k$ and $m$ are constants, find the possible values of $k$ and $m$

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If the simultaneous linear equations in $x$ and $g$ , $y = kx + m$ and $y= (2k - 1)x +4$ have at least one solution, where $k$ and $m$ are constants, find the possible values of $k$ and $m$