Which Equations Are Linear in Variables x, y, and z?

[4Fun]

Homework Statement

Let k \in \Re / {0}. Which of the three equations is linear in x, y, z?

x + y - z = tan(k)

kx - (1/k)y = 6

(3^k)x + y + z = 12


Well the question is basic, but I'm a little stumped because the questions asks for a single linear equation. By definition a linear equation is linear if it contains only terms that are constants or products of constants and variables in the first power. By that definition I think all should be linear in x, y and z. E.g. since k is a constant, tan(k) would also be a constant, right? The second equation is also linear, but is it linear in all three variables? I think it is linear in all three since it basically also includes the product of z and the constant 0.
And I think the third equation is similar as well, if k is supposed to be a constant.

Can anybody just quickly tell me if there really is only one linear equation here and whether what I have come up with is correct?

 

[Mark44]

Homework Statement

Let k \in \Re / {0}. Which of the three equations is linear in x, y, z?

x + y - z = tan(k)

kx - (1/k)y = 6

(3^k)x + y + z = 12


Well the question is basic, but I'm a little stumped because the questions asks for a single linear equation. By definition a linear equation is linear if it contains only terms that are constants or products of constants and variables in the first power. By that definition I think all should be linear in x, y and z. E.g. since k is a constant, tan(k) would also be a constant, right? The second equation is also linear, but is it linear in all three variables? I think it is linear in all three since it basically also includes the product of z and the constant 0.
And I think the third equation is similar as well, if k is supposed to be a constant.

Can anybody just quickly tell me if there really is only one linear equation here and whether what I have come up with is correct?

All three are linear. All three represent planes in R³. The fact that the equations include tan(k), 1/k, and 3^k respectively, has no bearing. As far as x, y, and z are concerned, k is a constant, so tan(k), 1/k, and 3^k are constants as well.

 

[4Fun]

Thanks a lot for the quick answer. One more question though, what does it mean for an equation to be linear in x, y and z?

 

[Ray Vickson]

Thanks a lot for the quick answer. One more question though, what does it mean for an equation to be linear in x, y and z?

You already explained that yourself in your previous response.

 

[Mentallic]

What if k=\pi/2+n\pi, n\in\mathbb{Z} ? We have a problem with \tan{k} in that situation.

 

[Mark44]

What if k=\pi/2+n\pi, n\in\mathbb{Z} ? We have a problem with \tan{k} in that situation.

Sure, tan(k) is undefined for these values, but I don't think that affects the "linearness" of the equation, with regard to x, y, and z.

 

[Mentallic]

Sure, tan(k) is undefined for these values, but I don't think that affects the "linearness" of the equation, with regard to x, y, and z.

Sure, I guess, but notice that they defined k\in\mathbb{R}/\{0\}
I'd hazard a guess that they excluded k=0 because of this equation

kx - (1/k)y = 6

But is it because of the kx term or the undefined y/k term? Or possibly both? If it's because of the undefined value, then I'd have to say that they probably just forgot to remove the values of k where tan(k) is undefined.

 

[Mark44]

kx - (1/k)y = 6

But is it because of the kx term or the undefined y/k term? Or possibly both? If it's because of the undefined value, then I'd have to say that they probably just forgot to remove the values of k where tan(k) is undefined.

That would be my guess as well. The problem writer might have gotten sloppy.