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Zorluk: OrtaSystems of Linear Equations

A freight logistics company charges fixed rates per unit for two types of cargo containers: Type XX and Type YY. Shipping 44 Type XX containers and 33 Type YY containers costs a total of $290\$290. Shipping 66 Type XX containers and 55 Type YY containers costs a total of $460\$460. Based on these rates, what is the total cost to ship 77 Type XX containers and 66 Type YY containers?

  1. A
    $375
  2. B
    $495
  3. $545Cevap
  4. D
    $630
  5. E
    $750

Cevap

$545
By solving the system 4x+3y=2904x + 3y = 290 and 6x+5y=4606x + 5y = 460, subtracting the first equation from the second yields 2x+2y=1702x + 2y = 170, so x+y=85x + y = 85. Adding x+y=85x + y = 85 to 6x+5y=4606x + 5y = 460 gives 7x+6y=5457x + 6y = 545. Alternatively, solving individually yields x=35x = 35 and y=50y = 50, giving 7(35)+6(50)=5457(35) + 6(50) = 545.

Adım Adım Çözüm

1
Set up a system of linear equations representing the given shipping costs.
Let xx be the cost per Type XX container and yy be the cost per Type YY container. Equation (1): 4x+3y=2904x + 3y = 290. Equation (2): 6x+5y=4606x + 5y = 460.
Translate the word problem statements into linear equations.
2
Subtract Equation (1) from Equation (2) to find a linear combination.
(6x + 5y) - (4x + 3y) = 460 - 290 \implies 2x + 2y = 170 \implies x + y = 85.
Determining the combined cost of one container of each type simplifies calculating larger combinations.
3
Add the expression for x+yx + y to Equation (2) to obtain 7x+6y7x + 6y.
(6x + 5y) + (x + y) = 460 + 85 \implies 7x + 6y = 545.
Combining 6x+5y6x + 5y and x+yx + y directly gives the cost for 77 Type XX and 66 Type YY containers without requiring individual variable solving.

Anahtar Kavram

Linear combinations and systems of linear equations in two variables
Tahmini Süre:1m 30s
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Konular

Systems of Linear Equations

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