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Difficulty: MediumSystems of Linear Equations

In a certain video game, players earn points by completing daily quests and weekly challenges. Last month, a player completed a total of 2424 activities, which consisted of daily quests and weekly challenges, and earned a total of 190190 points. Each daily quest completed was worth 55 points, and each weekly challenge completed was worth 1515 points. How many weekly challenges did the player complete?

  1. A
    55
  2. 77Answer
  3. C
    99
  4. D
    1717

Answer

The player completed 77 weekly challenges.
To find the number of weekly challenges, we set up a system of linear equations. Let dd be the number of daily quests and ww be the number of weekly challenges. We have d+w=24d + w = 24 and 5d+15w=1905d + 15w = 190. Expressing dd as 24−w24 - w and substituting it into the second equation gives 5(24−w)+15w=1905(24 - w) + 15w = 190, which simplifies to 120−5w+15w=190120 - 5w + 15w = 190. Combining like terms yields 120+10w=190120 + 10w = 190, so 10w=7010w = 70, which gives w=7w = 7. This matches the correct value.

Step-by-Step Solution

1
Represent the situation with a system of linear equations.
Let dd represent the number of daily quests completed and ww represent the number of weekly challenges completed. The system is:
d+w=245d+15w=190\begin{aligned} d + w &= 24 \\ 5d + 15w &= 190 \end{aligned}
This sets up the mathematical model using variables for the two unknown quantities.
2
Express one variable in terms of the other using the first equation.
d=24−wd = 24 - w
This allows for substitution into the second equation to reduce the system to a single variable.
3
Substitute the expression into the second equation and expand.
5(24−w)+15w=190  ⟹  120−5w+15w=1905(24 - w) + 15w = 190 \implies 120 - 5w + 15w = 190
This substitutes the representation of dd to solve for ww directly.
4
Simplify the equation and solve for ww.
120+10w=190  ⟹  10w=70  ⟹  w=7120 + 10w = 190 \implies 10w = 70 \implies w = 7
Combining like terms and isolating the variable yields the number of weekly challenges.

Key Concept

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

Alternative Method

Instead of substitution, the elimination method can be used. Multiply the first equation, d+w=24d + w = 24, by −5-5 to get −5d−5w=−120-5d - 5w = -120. Add this equation to the second equation, 5d+15w=1905d + 15w = 190, to eliminate dd, resulting in 10w=7010w = 70, which simplifies to w=7w = 7.
Estimated Time:1m 30s
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Topics

Systems of Linear Equations

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