Final answer:
To graph the level curves of the given functions, we need to choose different values for z and solve for x and y. For the function z = 3x + 2y, we find the equations of three level curves for z values of 0, 2, and 4. For the function z = 3x^2 + 9y, we find the equations of three level curves for z values of 0, 10, and 20.
Step-by-step explanation:
When graphing level curves of a function, we need to fix the value of z and find the corresponding values of x and y that satisfy the equation.
For the function z = 3x + 2y, let's choose z values of 0, 2, and 4. When z = 0, we have 3x + 2y = 0. By solving for y, we get y = -1.5x. Similarly, for z = 2 and z = 4, we get y = -1.5x - 1 and y = -1.5x - 2, respectively.
For the function z = 3x^2 + 9y, let's choose z values of 0, 10, and 20. When z = 0, we have 3x^2 + 9y = 0. By solving for y, we get y = -1/3x^2. Similarly, for z = 10 and z = 20, we get y = -1/3x^2 + 10/9 and y = -1/3x^2 + 20/9, respectively.