Simultaneous equations in context

Applying simultaneous equations to comparing plans, break-even analysis, mixtures and other worded problems.

Comparing plans

Which plan is cheaper?

Hover to compare. Below 100 minutes Plan B is cheaper; above it, Plan A.

Example 1●●

Plan A costs $\$30$ plus $\$0.10$ per minute; Plan B costs $\$15$ plus $\$0.25$ per minute. When do they cost the same?

$30 + 0.1m = 15 + 0.25m$

$15 = 0.15m$, so $m = 100$ minutes.

Example 2●●

A business has costs $C = 5000 + 12n$ and revenue $R = 20n$ for $n$ items. Find the break-even point.

$20n = 5000 + 12n$, so $8n = 5000$.

$n = 625$ items

Flashcards

Break-even

Revenue $=$ cost

Point of intersection means

Both conditions true at once

Profit

Revenue $-$ cost

Practice

Plan A: $\$30 + \$0.10$/min. Plan B: $\$15 + \$0.25$/min. Equal at how many minutes?
$C = 5000 + 12n$, $R = 20n$. Break-even $n$?
$2$ adults + $3$ children cost $\$55$; $3$ adults + $2$ children cost $\$65$. Enter the adult price, then the child price.
Using $C = 5000 + 12n$, $R = 20n$: profit when $n = 825$?
For $150$ minutes, which plan in Q1 is cheaper?
A $50$ L mix of two juices costs $\$110$. Juice A costs $\$2.50$/L and juice B $\$1.75$/L. Enter litres of A, then B.
A rectangle has perimeter $40$ and length $4$ more than width. Enter length, then width.
A taxi charges $\$4 + \$2$/km; a rideshare $\$10 + \$1.70$/km. At how many km are they equal?