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Zorluk: OrtaSimultaneous Equations and Systems

An event management firm hires two types of security personnel: Senior Officers and Junior Officers. Contract A requires 4 Senior Officers and 6 Junior Officers for a total daily cost of 2,180.ContractBrequires7SeniorOfficersand3JuniorOfficersforatotaldailycostof2,180. Contract B requires 7 Senior Officers and 3 Junior Officers for a total daily cost of 2,690. Based on this system of equations, match each quantity on the left with its correct dollar value on the right.

  • Daily rate of one Senior Officer$320
  • Daily rate of one Junior Officer$150
  • Difference in daily rates between one Senior Officer and one Junior Officer$170
  • Combined daily rate for a team of 1 Senior Officer and 1 Junior Officer$470

Cevap

Senior Officer rate = 320;JuniorOfficerrate=320; Junior Officer rate = 150; Difference in rates = 170;Combinedrate=170; Combined rate = 470.
Solving the linear system yields s=320s = 320 for Senior Officers and j=150j = 150 for Junior Officers. The difference between their rates is 320−150=170,andtheircombineddailyrateis320 - 150 = 170, and their combined daily rate is 320 + 150 = 470.

Adım Adım Çözüm

1
Formulate a system of linear equations from the given contract conditions.
Let ss be the daily rate of a Senior Officer and jj be the daily rate of a Junior Officer. Equation 1: 4s+6j=21804s + 6j = 2180. Equation 2: 7s+3j=26907s + 3j = 2690.
Translating context into algebraic form allows systematic elimination.
2
Eliminate the variable jj to solve for ss.
Divide Equation 1 by 2: 2s+3j=10902s + 3j = 1090. Subtract this from Equation 2 (7s+3j=26907s + 3j = 2690): (7s+3j)−(2s+3j)=2690−1090  ⟹  5s=1600  ⟹  s=320(7s + 3j) - (2s + 3j) = 2690 - 1090 \implies 5s = 1600 \implies s = 320.
Matching coefficients of jj isolates variable ss cleanly.
3
Substitute s=320s = 320 into 2s+3j=10902s + 3j = 1090 to calculate jj.
2(320)+3j=1090  ⟹  640+3j=1090  ⟹  3j=450  ⟹  j=1502(320) + 3j = 1090 \implies 640 + 3j = 1090 \implies 3j = 450 \implies j = 150.
Determines the second variable value in the system.
4
Compute the difference (s−js - j) and sum (s+js + j) of the two rates.
Difference: 320−150=170320 - 150 = 170. Combined rate: 320+150=470320 + 150 = 470.
Provides the exact target values for all items in the matching task.

Anahtar Kavram

Solving simultaneous linear equations by variable elimination and substitution
Tahmini Süre:2m 0s
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Konular

Simultaneous Equations and Systems

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