Basic Linear equation word problems

Start with real numbers. A gym costs $10$ to join, then $5$ for each month. Work out a few totals and watch what happens.

One month: $10 + 5 = 15$. Two months: $10 + 5 + 5 = 20$. Three months: $10 + 5 + 5 + 5 = 25$. The $10$ never changes; each extra month just adds one more $5$.

So for $m$ months you add $5$ a total of $m$ times: the cost is $10 + 5m$. That is the whole idea: find the part that stays fixed, and the part that repeats once per unit. Written generally this is $y = mx + c$, where $c$ is the fixed amount ($10$ here) and $m$ is the amount per unit ($5$ here).

If the quantity goes down instead of up, the per-unit part is subtracted. A candle $10$ inches tall that burns $0.5$ inches an hour is $9.5$ after one hour, $9$ after two, and $10 - 0.5t$ after $t$ hours.

To answer a question, put in the given number, or set the expression equal to a target. Profit is revenue minus cost; and to find when two changing amounts are equal, set their two expressions equal.

A taxi costs $3$ to start, then $2$ per mile. One mile: $3 + 2 = 5$. Two miles: $3 + 2 + 2 = 7$. Three miles: $3 + 2 + 2 + 2 = 9$. So $m$ miles cost $3 + 2m$.

A candle is $10$ inches tall and burns $0.5$ inches each hour. After one hour it is $9.5$, after two it is $9$. Since it shrinks, the rate is subtracted: after $t$ hours the height is $h = 10 - 0.5t$.

Country A starts at $12$ and adds $0.42$ per year, so $E = 12 + 0.42t$. Country B starts at $18.4$ and adds $0.28$ per year, so $E = 18.4 + 0.28t$. Setting $12 + 0.42t = 18.4 + 0.28t$ finds the year $t$ when the two are equal.