The original Going Rate measured one thing changing at a time. Out in the real world, things change against each other: two savers, two runners, two plans competing for your money. Reading that takes a system, a pair of rate equations working together. This is where rates meet algebra.
The Going Rate taught you that a rate is amount ÷ time. Here's the upgrade: when you graph an amount that changes at a constant rate, you always get a straight line, and the rate is the line's slope. A starting amount that isn't zero just slides the line up or down. In function form: y = mx + b, where m is the rate of change and b is the starting value.
Now put two rate-lines on the same graph. Where they cross is the one moment both stories agree, same amount, same time. That crossing point is the solution to a system of equations, and you can find it two ways: read the graph, or solve it with algebra by setting the two expressions for y equal to each other.
There's a faster way to solve some systems: combine the two rates into one relative rate before you even touch the distance. If two things move toward each other, add the speeds; the gap closes at their sum. If one is chasing another, subtract the speeds; the gap closes at their difference. This is really the same algebra as Part 2, just done by elimination instead of substitution.
Scientists rarely get a rate in the unit they need. The fix is a chain of conversion factors, each one secretly equal to 1 (like 1609 m / 1 mi), stacked so the unwanted units cancel diagonally and only the target unit survives.
To honestly compare a mountain eroding with a beam of light, you need everything in the same unit. Converted to meters per second, real-world rates span roughly 20 orders of magnitude, from 10-12 to 108. Scientific notation is how you write, and compare, numbers that extreme without drowning in zeros.
Every rate so far has added the same amount each period, that's linear growth: a constant difference. But some real rates multiply by the same percentage each period instead, a constant ratio. That's exponential growth, and it eventually beats any linear rate, no matter how big a head start the linear one gets.
| Period | Linear | Exponential |
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Systems of equations aren't just a graphing exercise, they're how you make real decisions. Every "which plan is cheaper" question is a system: one line per plan, and the break-even point tells you exactly when the winner changes.
Ten questions on systems, relative rate, dimensional analysis, magnitude, and growth type. Set up the equations before you guess.
A single rate tells you how fast one thing changes. The moment a second rate enters the picture, savings vs. spending, one runner vs. another, one plan vs. another, you're solving a system, and the crossing point is the answer that matters. Substitution, elimination, relative rate: three names for the same idea, finding the instant two stories say the same thing.