In a certain video game, players earn points for completing quests and defeating bosses. C… — SAT

In a certain video game, players earn points for completing quests and defeating bosses. Completing a quest earns $q$ points, and defeating a boss earns $b$ points. Leo completed $5$ quests and defeated $3$ bosses, earning a total of $250$ points. Maya completed $7$ quests and defeated $2$ bosses, earning a total of $240$ points. What is the value of $q$ ?

Answer: 20

Answer

The value of

q

20

To find the value of

q

, we translate the given scenarios into a system of two linear equations:

5q + 3b = 250

and

7q + 2b = 240

. Multiplying the first equation by

2

gives

10q + 6b = 500

. Multiplying the second equation by

3

gives

21q + 6b = 720

. Subtracting the first equation from the second yields

(21q - 10q) + (6b - 6b) = 720 - 500

, which simplifies to

11q = 220

. Dividing both sides by

11

results in

q = 20

Step-by-Step Solution

Set up the system of linear equations from the given information.

5q + 3b = 250

and

7q + 2b = 240

To represent the points earned by Leo and Maya mathematically.

Multiply the first equation by

2

and the second equation by

3

10q + 6b = 500

and

21q + 6b = 720

To make the coefficients of

b

equal so they can be eliminated.

Subtract the first modified equation from the second modified equation.

11q = 220

To eliminate

b

and solve for

q

directly.

Divide both sides of the equation by

11

q = 20

To find the number of points earned per completed quest.

Key Concept

Solving systems of two linear equations in two variables using elimination.

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Answer

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Key Concept

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