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Difficulty: HardSystems of Linear Equations

A software analytics company offers three annual subscription plans: Standard, Professional, and Enterprise.

• A client purchasing 3 Standard, 2 Professional, and 1 Enterprise plan pays a total of 1,110.���Aclientpurchasing1Standard,4Professional,and2Enterpriseplanspaysatotalof1,110. ��� A client purchasing 1 Standard, 4 Professional, and 2 Enterprise plans pays a total of 1,620.
• A client purchasing 4 Standard, 1 Professional, and 3 Enterprise plans pays a total of $1,730.

What is the cost, in dollars, of 1 Enterprise plan?

Answer: 350 dollars

Answer

The cost of 1 Enterprise plan is 350 dollars.
Setting up equations for each purchase tier gives 3x+2y+z=11103x + 2y + z = 1110, x+4y+2z=1620x + 4y + 2z = 1620, and 4x+y+3z=17304x + y + 3z = 1730. Isolating xx in the second equation gives x=1620−4y−2zx = 1620 - 4y - 2z. Substituting xx into the first and third equations yields 2y+z=7502y + z = 750 and 3y+z=9503y + z = 950, respectively. Subtracting these reduced equations gives y=200y = 200, which upon back-substitution into 2y+z=7502y + z = 750 reveals z=350z = 350. Thus, 1 Enterprise plan costs 350 dollars.

Step-by-Step Solution

1
Formulate the linear system of equations from the given conditions.
Let xx be the price of a Standard plan, yy the price of a Professional plan, and zz the price of an Enterprise plan:
(1) 3x+2y+z=11103x + 2y + z = 1110
(2) x+4y+2z=1620x + 4y + 2z = 1620
(3) 4x+y+3z=17304x + y + 3z = 1730
Translating the verbal conditions into algebraic equations creates a solvable system.
2
Isolate variable xx in equation (2) and substitute it into equations (1) and (3).
From (2), x=1620−4y−2zx = 1620 - 4y - 2z.
Substituting into (1):
3(1620−4y−2z)+2y+z=1110  ⟹  4860−12y−6z+2y+z=1110  ⟹  10y+5z=3750  ⟹  2y+z=7503(1620 - 4y - 2z) + 2y + z = 1110 \implies 4860 - 12y - 6z + 2y + z = 1110 \implies 10y + 5z = 3750 \implies 2y + z = 750 (Equation 4)

Substituting into (3):
4(1620−4y−2z)+y+3z=1730  ⟹  6480−16y−8z+y+3z=1730  ⟹  15y+5z=4750  ⟹  3y+z=9504(1620 - 4y - 2z) + y + 3z = 1730 \implies 6480 - 16y - 8z + y + 3z = 1730 \implies 15y + 5z = 4750 \implies 3y + z = 950 (Equation 5)
Eliminating xx reduces the system to two linear equations with two variables.
3
Solve the 2x2 system of equations for yy and zz.
Subtracting Equation (4) from Equation (5):
(3y+z)−(2y+z)=950−750  ⟹  y=200(3y + z) - (2y + z) = 950 - 750 \implies y = 200.

Substitute y=200y = 200 back into Equation (4):
2(200)+z=750  ⟹  400+z=750  ⟹  z=3502(200) + z = 750 \implies 400 + z = 750 \implies z = 350.
Solving the reduced system yields the exact values of yy and zz.

Key Concept

Solving Systems of Three Linear Equations via Gaussian Elimination / Variable Substitution
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Topics

Systems of Linear Equations

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