What is the break-even point?

The amount at which two plans, or a cost and an income, are exactly equal. On a graph it is the point where the two lines cross.

How do you find the break-even point of two plans?

Write each cost as y = mx + b, set the two expressions equal, and solve for x. Then put x into either one to find the cost.

How do you know which plan is cheaper?

Try a number on each side of the break-even point and compare the two costs. The plan with the lower cost on that side is the cheaper one there.

Can two plans never break even?

Yes. If both plans charge the same price per unit, their lines are parallel and never cross, so one plan is always cheaper.

Part 18 of 20 · EquationsSee the whole course

How to find the break-even point (which plan is cheaper)

By The Pip team · October 23, 2026 · 6 min read

Short answer

Write each cost as a line and find where the lines cross. That crossing is the break-even point: the amount where both plans cost the same. Below it one plan is cheaper, above it the other is.

Write each plan as a line

A plan usually has a starting price and a price per unit. The per-unit price is the slope m. The starting price is b. Suppose, with x as the gigabytes (GB) used:

Plan A: 10 to start, plus 1 per GB, so y = x + 10
Plan B: 4 to start, plus 3 per GB, so y = 3x + 4

Find the break-even point

Both plans cost the same when the two right-hand sides are equal:

x + 10 = 3x + 4
6 = 2x
x = 3 GB
Cost: 3 + 10 = 13, and 3(3) + 4 = 13 ✓

The break-even point is (3, 13): at 3 GB both plans cost 13. Move the slider below and watch the cheaper plan change at exactly that point.

Plan B is cheaper by 2

Plan A: 10 + 1x
Plan B: 4 + 3x

Data used (GB) 2

A: 10 + 1(2) = 12
B: 4 + 3(2) = 10

At 2 GB, plan A costs 12 and plan B costs 10. Plan B is cheaper here. The plans break even at 3 GB, where both cost 13.

Which plan is cheaper?

Do not guess from the starting prices. Try a number on each side of the break-even:

At 2 GB: A = 2 + 10 = 12 and B = 3(2) + 4 = 10, so B is cheaper.
At 4 GB: A = 4 + 10 = 14 and B = 3(4) + 4 = 16, so A is cheaper.

So B is cheaper for less than 3 GB, and A is cheaper for more than 3 GB. The plan with the lower starting price wins at first. The plan with the lower rate per GB wins in the long run.

A second example

Plan A is 15 + 2x and plan B is 5 + 4x. Set them equal: 15 + 2x = 5 + 4x, so 10 = 2x and x = 5 GB. The cost there is 15 + 2(5) = 25. Switch the figure to this pair to see it.

Watch it explained in under a minute

The same idea, narrated step by step. Square version for sharing: download the 1:1 video.

The five mistakes to avoid

Comparing only the starting prices. The cheaper-to-start plan stops being cheaper once you use enough.
Giving the cost when the question asks for the amount. The break-even has two numbers: 3 GB and a cost of 13.
Mixing up which side is which. Test a number on each side of the crossing.
Assuming the plans always cross. Two plans with the same price per GB are parallel and never break even.
Leaving out the units. Say “3 GB” and “13”, not just 3.

Try it yourself

Tap a question to see the answer.

1. Gym X costs 30 to join and 5 per visit. Gym Y has no joining fee and costs 10 per visit. When do they cost the same?

30 + 5x = 10x, so 30 = 5x and x = 6 visits. Both cost 60. Y is cheaper before 6 visits, X after.

2. Plan A is 20 + 2x and plan B is 8 + 5x. Where do they break even?

20 + 2x = 8 + 5x, so 12 = 3x and x = 4. The cost is 20 + 2(4) = 28.

3. Taxi P charges 2 + 3 per km. Taxi Q charges 5 + 2 per km. For which trips is P cheaper?

2 + 3x = 5 + 2x gives x = 3 km (both cost 11). Try 1 km: P = 5 and Q = 7, so P is cheaper for trips shorter than 3 km.

Where this fits in

“Cheaper than” is really an inequality, which is where this course goes next. Start with how to solve a linear inequality.