Problema di sistema lineare per noleggio videogiochi

Problem

You rented $6$ video games. New games cost \4$each, and older games cost$$2$each. If the total cost was$$22$, how many new games and older games did you rent?

Step 1: Name the unknown quantities

Let the number of new games be $n$, and let the number of older games be $o$.

Step 2: Write the total-count equation

The new games and older games together make $6$ rentals, so

$$n+o=6.$$

Step 3: Write the total-cost equation

Each new game costs \4$, and each older game costs$$2$. Since the total cost is$$22$,

$$4n+2o=22.$$

Step 4: Substitute for one variable

From the total-count equation,

$$n+o=6,$$

solve for $o$:

$$o=6-n.$$

Step 5: Solve the cost equation

Substitute $6-n$ for $o$ in the cost equation:

$$4n+2(6-n)=22.$$

Simplify:

$$4n+12-2n=22,$$ $$2n+12=22,$$ $$2n=10,$$ $$n=5.$$

Step 6: Find the other quantity

Use the total-count equation:

$$n+o=6.$$

Since $n=5$,

$$5+o=6,$$

so

$$o=1.$$

Step 7: State the answer

You rented $5$ new games and $1$ older game.