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SAGeneral MathematicsSyllabus dot point

How do we find the point where two linear models meet, and what does break-even mean?

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

How to solve two linear equations by substitution or elimination, find the break-even point where cost equals revenue, and interpret profit and loss regions.

Generated by Claude Opus 4.76 min answer

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  1. What this dot point is asking
  2. Solving simultaneous equations
  3. Break-even analysis
  4. Profit, loss, and interpretation

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.

Break-even analysis

A business has a cost function and a revenue function:

C=(variable cost per item) x+(fixed costs),R=(selling price) xC = (\text{variable cost per item})\,x + (\text{fixed costs}), \qquad R = (\text{selling price})\,x

Profit, loss, and interpretation

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

P=R−CP = R - C

For the stall, P=7x−(3x+200)=4x−200P = 7x - (3x + 200) = 4x - 200. Each extra item adds 4ofprofit(thegradientof4 of profit (the gradient of P).Thegraphshowsrevenueandcostlinescrossingat). The graph shows revenue and cost lines crossing at x = 50$; to the left is the loss region, to the right the profit region.