Question

Describe the level curves of the function. Sketch the level curves for the given \(c\) -values. $$ z=6-2 x-3 y, \quad c=0,2,4,6,8,10 $$

Short answer

The level curves for the function \(z = 6 - 2x - 3y\) are the lines obtained by rearranging \(z = c\) to obtain \(y = (2x - c + 6) / -3\) for each \(c\) - value. For example, for \(c = 0\), the level curve is \(y = (2x + 6) / -3\); for \(c = 2\), it is \(y = (2x + 4) / -3\), and so on until \(c = 10\), where the curve is \(y = (2x - 4) / -3\).

Step-by-step solution

Understanding level curves

Level curves are two-dimensional graphs that represent slices of three-dimensional graphs. For a function \(z = f(x,y)\), the level curve for a given value \(c\) is the graph of the equation \(f(x,y) = c\) in the \(xy\) - plane.

Expressing the function in terms of c

We have the function \(z = 6 - 2x - 3y\). We need to express this function in terms of \(c\). So we set \(z = c\) to obtain the equation: \(c = 6 - 2x - 3y\). We can then rearrange this to isolate \(y\), giving: \(y = (2x - c + 6) / -3\).

Plotting the level curves for given c - values

Now we need to sketch the level curves for the given \(c\) - values. We do this by substituting each \(c\) - value into the equation we just found and then sketching the resulting line in the \(xy\) - plane. For \(c = 0\), we get \(y = (2x + 6) / -3\), for \(c = 2\), we get \(y = (2x + 4) / -3\), and so on, until \(c = 10\), for which we get \(y = (2x - 4) / -3\). Each of these can be hence drawn on an \(xy\) - plane.