Finding the Value of k for No Solution in a System of Equations

[GLprincess02]

Another question on my homwork reads: "For what value of k does
$$\left\{\begin{array}Kx+2y=12\\9x+2y=8\end{array}\right$$ have no solution? Justify."


Is the answer 9 or -9, or neither? I don't understand it, nor can I justify it.

 

[GLprincess02]

Sorry, there should be a k in front of the x in the 1st equation.

 

[cristo]

Try subtracting the 2nd eqn from the 1st. Does this help?

 

[GLprincess02]

So then it would be 9? Because it would then be 0 = 4, and that would make it inconsistent?

 

[cristo]

Yes, or to put it another way (k-9)x=4 has no solution for k=9, since division by 0 is not defined.

 

[GLprincess02]

Ok that makes sense. But one more thing...how can you determine whether to subtract the 2nd equation from the 1st, or add them both together?

 

[cristo]

Because 2y appears in both equations. Subtracting makes them cancel out so you can get an expression for x.

In fact, you don't have to subtract at all, you can just spot that putting k=9 will make the LHS of both equations the same, whereas the RHS of both are different. This shows inconsistency in the set of equations.

However, I just suggested subtracting, since this would enable you to spot the answer more quickly!

 

[GLprincess02]

Ohhhh ok it all makes sense now. Thank you very much for all your help.