Why do I need systems of equations to pick a phone plan?

Slide your data use back and forth. Watch which plan's line is lower where you are.

Challenge: Set your data use right at the break-even point. The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

Each plan's cost is a linear equation: $C = b + mx$. The fee $b$ is where the line starts on the cost axis, and the price per GB $m$ is how steeply it climbs.

Two plans make a system: two equations that must both be true at the break-even point. Set them equal and solve for $x$:

$$b_A + m_A x = b_B + m_B x \;\Rightarrow\; x = \frac{b_B - b_A}{m_A - m_B}$$

Look at the denominator. If $m_A = m_B$, it is zero. You can't divide by zero, and the graph shows why: the lines are parallel and never meet. That is a real, useful answer (one plan is always cheaper), so the site shows it with the code #NUM! and still counts the puzzle as solved.

Any "fixed cost plus per-unit cost" comparison works the same way: rentals by the mile, gym fees per visit, subscriptions versus pay-as-you-go, or leasing versus buying equipment. Enter each option's fixed part as the fee and its per-unit part as the rate.

• Use your typical month, not your worst month, for data use.
• Taxes and fees that scale with price don't change which plan is cheaper, but they do change the dollar amounts.