Solve pairs of simultaneous linear equations algebraically and graphically, and interpret the solution as a break-even point in cost and revenue models.

What this dot point is asking

You must solve a pair of linear equations and interpret the intersection, especially as a break-even quantity in a business model.

Solving simultaneous equations

The solution is the point where the two lines cross. There are three methods.

Substitution

Rearrange one equation for a variable and substitute into the other.

Elimination

Add or subtract multiples of the equations to cancel one variable.

Graphical

Plot both lines; the intersection is the solution.

Each method has its place. Substitution is cleanest when one equation already has a variable by itself, such as $y = 2x + 1$. Elimination is fastest when the same variable has matching or opposite coefficients, so adding or subtracting cancels it outright. The graphical method is the one to reach for when a question supplies a graph or asks you to read the intersection visually, though it gives only an approximate answer unless the crossing point lands on grid lines.

Break-even analysis

A business has a cost function and a revenue function:

Profit, loss, and interpretation

Profit is the vertical gap between the revenue and cost lines:

For the stall, $P = 7x - (3x + 200) = 4x - 200$. Each extra item adds 4 dollars of profit (the gradient of $P$). The graph shows revenue and cost lines crossing at $x = 50$; to the left is the loss region, to the right the profit region.

Interpreting break-even in context

The break-even quantity answers the practical question "how many must we sell before we stop losing money?" Below it, total cost exceeds revenue and the venture runs at a loss; above it, every additional sale contributes to profit at a rate equal to the difference between the selling price and the variable cost per item. This per-item margin is exactly the gradient of the profit line. A common follow-up question asks how many items are needed to reach a target profit: set the profit function equal to the target and solve, rather than re-deriving from cost and revenue separately.

When the model has no break-even

If the selling price per item is less than or equal to the variable cost per item, the revenue line is never steeper than the cost line, so the two never cross for positive output and the business can never break even. Recognising this from the gradients, before attempting to solve, is a useful check: a break-even point only exists when the price exceeds the variable cost so that the profit gradient is positive.