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anonymous
 5 years ago
Which values of constant k does the system of linear equations give exactly one solution (x1)+(4x)(2)=1 and (3)(x1) + (12)(x2) = k
anonymous
 5 years ago
Which values of constant k does the system of linear equations give exactly one solution (x1)+(4x)(2)=1 and (3)(x1) + (12)(x2) = k

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Is this the system you are trying to solve? $$x_1 + 4x_2 = 1$$ $$3x_1 + 12 x_2 = k$$

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Is it any number except 3?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0because if its 3 then there will be infinite solutions

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0If so, the answer is that there is no such value. There is one value for which there is an infinite number of solutions, and an infinite number of values for which there are no solutions. To prove it, consider that two equations have a unique solution if and only if the determinant of the coefficient matrix is zero. But the determinant of the coefficient matrix in this case is \(12\times1  3\times 4 = 0\). Note that \(k\) plays no part in it, so no amount of tweaking \(k\) can change the determinant.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0What do you mean by determinant of the coefficient matrix?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0The coefficients infront of each x?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yes. The determinant of a matrix is a complicated beast, but for a 2x2 matrix is as below: $$ \det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad  bc $$

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0doesnt the answer of each equation go into the matrix as well?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0like for the equations shouldn't it be like 1 4 1 3 12 k

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0sorry I'm new to matrix

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yes, but the coefficient matrix is the one that is made up by just the coefficients of the variables. So, in this case, it's: $$\begin{bmatrix}1 & 4 \\ 3 & 12\end{bmatrix}$$
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