Break down

Costs and Sales can be broken down, which provide further insight into operations.
Decomposing Total Costs as Fixed Costs plus Variable Costs.

One can decompose Total Costs as Fixed Costs plus Variable Costs:

\text{TC} = \text{TFC} + \text{V} \times \text{X}

Decomposing Sales as Contribution plus Variable Costs.

Following a matching principle of matching a portion of sales against variable costs, one can decompose Sales as Contribution plus Variable Costs, where contribution is "what's left after deducting variable costs". One can think of contribution as "the marginal contribution of a unit to the profit", or "contribution towards offsetting fixed costs".

In symbols:

\begin{align} \text{TR} &= \text{P} \times \text{X}\\ &= \bigl(\left(\text{P} - \text{V} \right)+\text{V}\bigr)\times \text{X}\\ &= \left(\text{C}+\text{V}\right)\times \text{X}\\ &= \text{C}\times\text{X} + \text{V}\times \text{X} \end{align}

where

C = Unit Contribution (Margin)

Profit and Loss as Contribution minus Fixed Costs.

Subtracting Variable Costs from both Costs and Sales yields the simplified diagram and equation for Profit and Loss.

In symbols:

\begin{align} \text{PL} &= \text{TR} - \text{TC}\\ &= \left(\text{C}+\text{V}\right)\times \text{X} - \left(\text{TFC} + \text{V} \times \text{X}\right)\\ &= \text{C} \times \text{X} - \text{TFC} \end{align}

Diagram relating all quantities in CVP.

These diagrams can be related by a rather busy diagram, which demonstrates how if one subtracts Variable Costs, the Sales and Total Costs lines shift down to become the Contribution and Fixed Costs lines. Note that the Profit and Loss for any given number of unit sales is the same, and in particular the break-even point is the same, whether one computes by Sales = Total Costs or as Contribution = Fixed Costs. Mathematically, the contribution graph is obtained from the sales graph by a shear, to be precise \left(\begin{smallmatrix}1 & 0\\ -V & 1\end{smallmatrix}\right), where V are Unit Variable Costs.