A venue offers two types of event packages: Standard and Deluxe. The total cost of 3 Stand… — GRE General Test

A venue offers two types of event packages: Standard and Deluxe. The total cost of 3 Standard packages and 2 Deluxe packages is $410. The total cost of 2 Standard packages and 5 Deluxe packages is$ 640. What is the cost, in dollars, of 1 Deluxe package?

Answer: 100 dollars

Answer

100

Setting up the linear system

3S + 2D = 410

and

2S + 5D = 640

allows us to multiply the equations by 2 and 3 respectively, obtaining

6S + 4D = 820

and

6S + 15D = 1920

. Subtracting the two equations eliminates

S

and gives

11D = 1100

, leading to

D = 100

Step-by-Step Solution

Define variables and establish the system of linear equations.

Let

S

represent the cost of a Standard package and

D

represent the cost of a Deluxe package.
Equation 1:

3S + 2D = 410

Equation 2:

2S + 5D = 640

Translating the scenario into mathematical equations forms a 2x2 system of linear equations.

Use elimination to eliminate variable

S

Multiply Equation 1 by 2:

6S + 4D = 820

Multiply Equation 2 by 3:

6S + 15D = 1920

Creating matching coefficients for

S

allows elimination by subtraction.

Subtract the transformed equations and solve for

D

(6S + 15D) - (6S + 4D) = 1920 - 820 \implies 11D = 1100 \implies D = 100

Subtracting cancels out

S

, leaving a single linear equation in terms of

D

Key Concept

Solving 2x2 Systems of Linear Equations via Elimination

Alternative Method

Use the substitution method: Solve for

S

in terms of

D

from the first equation (

S = \frac{410 - 2D}{3}

) and substitute this into the second equation (

2\left(\frac{410 - 2D}{3}\right) + 5D = 640

). Multiplying both sides by 3 yields

820 - 4D + 15D = 1920

, which simplifies to

11D = 1100

, so

D = 100

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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