Systems of equations for video game purchases

A store sells used and new video games.  New video games cost more than used video games.  All used video games cost the same.  All new video games cost the same.

Omar spent a total of $84 on 4 used video games and 2 new video games.  Sally spent a total of $78 on 6 used video games and 1 new video game.  Janet has $120 to spend.

Find the number of used video games Janet can purchase after she purchases 3 new video games.

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n = price of new video game
u = price of used video game

4u + 2n = 84   {Omar spent a total of $84 on 4 used video games and 2 new video games}
6u + n = 78   {Sally spent a total of $78 on 6 used and 1 new video game}

Solve the system by the elimination method.

4u + 2n = 84   {top equation stays the same}
-12u - 2n = -156   {multiplied bottom equation by -2}
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-8u = -72   {added the two equations}
u = 9   {divided each side by 9}

Price of used game is $9.

6u + n = 78   {bottom original equation}
6(9) + n = 78   {substituted 9, in for u, into bottom original equation}
54 + n = 78   {multiplied 6 by 9}
n = 24   {subtracted 54 from each side}

Price of new game is $24.

How many used games can Janet purchase, with $120, after she purchases 3 new video games?

Let x = the number of used games Janet can purchase

3(24) + 9x = 120    {added 3 new games at $24 each with an unknown number of used games at $9 each}
72 + 9x = 120   {multiplied 3 by 24}
9x = 48   {subtracted 72 from each side}
x ≈ 5.333...   {divided each side by 9}

She can purchase 5 used games, after purchasing 3 new games, all with $120.

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