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Difficulty: MediumSystems of Linear Equations

A corporate catering service offers two lunch options: a Standard Meal and a Premium Meal. On Monday, an order of 4040 Standard Meals and 2020 Premium Meals cost a total of $1,300\$1,300. On Tuesday, an order of 2020 Standard Meals and 3030 Premium Meals cost a total of $1,250\$1,250. What is the total cost of an order consisting of 1515 Standard Meals and 1515 Premium Meals?

  1. A
    $650.00\$650.00
  2. $712.50\$712.50Answer
  3. C
    $812.50\$812.50
  4. D
    $975.00\$975.00
  5. E
    $1,162.50\$1,162.50

Answer

The total cost of 15 Standard Meals and 15 Premium Meals is $712.50.
By setting up the system 40x+20y=130040x + 20y = 1300 and 20x+30y=125020x + 30y = 1250, we simplify to 2x+y=652x + y = 65 and 2x+3y=1252x + 3y = 125. Subtracting the first from the second gives 2y=602y = 60, so y=30y = 30. Substituting y=30y = 30 into 2x+y=652x + y = 65 yields 2x=352x = 35, so x=17.50x = 17.50. The sum of one Standard Meal and one Premium Meal is x+y=47.50x + y = 47.50. Therefore, 1515 of each costs 15×47.50=$712.5015 \times 47.50 = \$712.50. Alternatively, adding the two simplified equations directly yields 4x+4y=190  ⟹  x+y=47.504x + 4y = 190 \implies x + y = 47.50, so 15(x+y)=15×47.50=$712.5015(x + y) = 15 \times 47.50 = \$712.50.

Step-by-Step Solution

1
Set up a system of linear equations using variables xx for the price of a Standard Meal and yy for the price of a Premium Meal.
40x+20y=130040x + 20y = 1300 and 20x+30y=125020x + 30y = 1250
Translate the scenario information into algebraic equations.
2
Simplify both equations by dividing by their greatest common factors.
Equation 1: 2x+y=652x + y = 65; Equation 2: 2x+3y=1252x + 3y = 125
Simplifying equations reduces computation complexity.
3
Subtract Equation 1 from Equation 2 to eliminate xx and solve for yy.
(2x+3y)−(2x+y)=125−65  ⟹  2y=60  ⟹  y=30(2x + 3y) - (2x + y) = 125 - 65 \implies 2y = 60 \implies y = 30
Elimination isolates variable yy.
4
Substitute y=30y = 30 back into Equation 1 to solve for xx.
2x+30=65  ⟹  2x=35  ⟹  x=17.502x + 30 = 65 \implies 2x = 35 \implies x = 17.50
Finding the value of xx gives the cost of one Standard Meal.
5
Calculate the target linear combination 15x+15y=15(x+y)15x + 15y = 15(x + y).
15(17.50+30.00)=15(47.50)=712.5015(17.50 + 30.00) = 15(47.50) = 712.50
Multiply the combined price of one of each meal by 15.

Key Concept

Solving systems of two linear equations in two variables using elimination and finding linear combinations.
Estimated Time:1m 45s
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Topics

Systems of Linear Equations

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