finding unknowns by creating systems of equations

Here you are asked to find unknown values, but you are not handed a ready-made equation to solve.
Instead you are given facts and relationships, and your job is to translate each one into an equation.
Every condition you are given becomes one equation.

Counts and totals are a common source of equations.
Suppose a jar holds $30$ coins, all nickels and dimes, worth $240$ cents.
"There are $30$ coins in all" becomes $n + d = 30$, and "they are worth $240$ cents" becomes $5n + 10d = 240$.

Relationships between the unknowns are equations too.
"There are twice as many dimes as nickels" becomes $d = 2n$, and "the two amounts add to $40$" becomes $x + y = 40$.
Any stated link between the unknowns can be written down as an equation.

Features of a graph work the same way.
A point the graph passes through, an intercept, or an asymptote is a condition you can substitute in to form an equation.
For example, a curve through $(0, 5)$ gives $5 = a(0)^2 + c$, so $c = 5$.

You need as many independent equations as there are unknowns.
Two unknowns need two equations; three unknowns need three.
Once you have enough, solve them together to find every value.

A theater sold $50$ tickets for a total of $\$380$. Adult tickets cost $\$10$ and child tickets cost $\$6$. Set up and solve a system to find how many of each.
Let $a$ be adult tickets and $c$ be child tickets.
The count gives $a + c = 50$, and the money gives $10a + 6c = 380$.
From the first, $a = 50 - c$; substituting gives $10(50 - c) + 6c = 380$, so $500 - 4c = 380$ and $c = 30$, $a = 20$.

A test has $20$ questions worth $100$ points in total. Multiple-choice questions are worth $4$ points each and essay questions $8$ points each. Set up a system for the number of each.
Let $m$ be multiple-choice and $e$ essay questions.
The count gives $m + e = 20$, and the points give $4m + 8e = 100$.
Substituting $m = 20 - e$ gives $4(20 - e) + 8e = 100$, so $80 + 4e = 100$ and $e = 5$, $m = 15$.

The function $y = a(2)^x + c$ has a horizontal asymptote at $2$ and a y-intercept at $1$. Form equations and find $a$.
The asymptote is a condition that gives $c = 2$.
The y-intercept is a condition at $x = 0$, giving $a(2)^0 + c = a + c = 1$.
Substituting $c = 2$ gives $a + 2 = 1$, so $a = -1$.