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Difficulty: HardSystems of Linear Equations

An event design company offers three distinct decorative bundles—Bundle X, Bundle Y, and Bundle Z—for corporate gala setups.

• Bundle X contains 3 floral arrangements, 2 LED uplights, and 1 table runner.
• Bundle Y contains 1 floral arrangement, 3 LED uplights, and 2 table runners.
• Bundle Z contains 2 floral arrangements, 1 LED uplight, and 3 table runners.

To decorate a venue, a coordinator orders a combination of these bundles containing a total of 26 floral arrangements, 23 LED uplights, and 23 table runners. If every bundle ordered is used in its entirety, what is the total number of bundles ordered by the coordinator?

  1. A
    10
  2. 12Answer
  3. C
    15
  4. D
    18
  5. E
    24

Answer

The total number of bundles ordered by the coordinator is 12.
Summing all three equations gives 6x+6y+6z=726x + 6y + 6z = 72. Factoring out 66 yields 6(x+y+z)=726(x + y + z) = 72, so dividing by 66 gives x+y+z=12x + y + z = 12. Alternatively, solving the system explicitly gives x=5x = 5, y=4y = 4, and z=3z = 3, whose sum is 5+4+3=125 + 4 + 3 = 12.

Step-by-Step Solution

1
Define variables and set up the system of linear equations based on item counts.
Let xx, yy, and zz represent the number of Bundle X, Bundle Y, and Bundle Z ordered, respectively.
3x+y+2z=26(Floral arrangements)2x+3y+z=23(LED uplights)x+2y+3z=23(Table runners)\begin{aligned} 3x + y + 2z &= 26 \quad \text{(Floral arrangements)} \\ 2x + 3y + z &= 23 \quad \text{(LED uplights)} \\ x + 2y + 3z &= 23 \quad \text{(Table runners)} \end{aligned}
Translate the word problem into a standard system of 3 linear equations in 3 variables.
2
Sum the three equations to find a direct linear combination for (x+y+z)(x + y + z).
(3x+2x+x)+(y+3y+2y)+(2z+z+3z)=26+23+23(3x + 2x + x) + (y + 3y + 2y) + (2z + z + 3z) = 26 + 23 + 23
6x+6y+6z=726x + 6y + 6z = 72
Notice that the sum of coefficients for each variable across all three equations is identical (3+2+1=63 + 2 + 1 = 6).
3
Factor out 6 and solve for the total number of bundles (x+y+z)(x + y + z).
6(x+y+z)=72  ⟹  x+y+z=126(x + y + z) = 72 \implies x + y + z = 12
Dividing both sides by 6 directly gives the required total quantity without needing to solve for individual variables xx, yy, and zz separately.

Key Concept

Linear combinations in systems of equations
Estimated Time:2m 0s
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Topics

Systems of Linear Equations

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