An entrepreneur has the following costs and revenue functions: TC = 30 + 3Q and TR = 4Q. Calculate the equilibrium quantity algebraically and graphically at the break-even point.

To find the equilibrium quantity algebraically at the break-even point, we need to equate total cost (TC) and total revenue (TR) and solve for the quantity (Q).

Given:
TC = 30 + 3Q
TR = 4Q

At the break-even point, the entrepreneur neither makes a profit nor incurs a loss. This means that the total revenue (TR) is equal to the total cost (TC). Therefore, we can set up the equation:

TR = TC

Replacing TR and TC with their respective formulas:

4Q = 30 + 3Q

Now, we can solve for Q:

4Q - 3Q = 30
Q = 30

The equilibrium quantity at the break-even point is Q = 30. This means that the entrepreneur needs to produce and sell 30 units of their product to break even.

To graphically visualize the break-even point, we can plot the total cost (TC) and total revenue (TR) functions on a graph. The break-even point will be where the two lines intersect.

First, let's rewrite the TC function as a linear equation in terms of Q:
TC = 30 + 3Q
TC = 3Q + 30

Now, plot the two functions on a graph using the quantity (Q) on the x-axis and cost (TC) and revenue (TR) on the y-axis:

- For the TC function, plot the point (0, 30) on the y-axis (intercept), and then use the slope of 3 to plot additional points. Connect the points to form a line.
- For the TR function, plot the point (0, 0) on the origin (no revenue at 0 quantity), and then use the slope of 4 to plot additional points. Connect the points to form a line.

The break-even point is where the two lines intersect, indicating the equilibrium quantity at which total revenue is equal to total cost. In this case, it will be the point (30, 120), where Q = 30 units and TC = TR = $120.

Note that the graphical approach can help visualize the break-even point, but the algebraic approach provides an exact solution.