NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Simultaneous equations and inequalities

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A printing company in Sydney offers two types of custom brochures. Brochure A costs $18 to produce and sells for $30, while Brochure B costs $24 to produce and sells for $42.

The company has a weekly production budget of $1800 and can produce up to 80 brochures per week. Let x be the number of Brochure A and y be the number of Brochure B.

The company wants to determine the exact number of Brochure A (x) that must be produced to exactly break even on total cost and total revenue, given that all brochures are sold. The system of equations is 18x + 24y = 1800 and 30x + 42y = 1800.

Solve exactly for x and provide the exact integer value.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The company must break even on total cost and total revenue, meaning total revenue equals total cost. The cost equation is 18x + 24y = 1800, and the revenue equation is 30x + 42y = 1800.
  2. Subtract the cost equation from the revenue equation to eliminate the constant term: (30x + 42y) - (18x + 24y) = 1800 - 1800.
  3. Simplify the left side: 30x - 18x = 12x, 42y - 24y = 18y. Right side: 0. So, 12x + 18y = 0.
  4. Solve 12x + 18y = 0 for x: 12x = -18y, so x = (-18y)/12 = (-3y)/2.
  5. Substitute x = -3y/2 into the cost equation: 18(-3y/2) + 24y = 1800.
  6. Calculate 18 * (-3y/2) = -54y/2 = -27y. So, -27y + 24y = 1800.
  7. Simplify: -3y = 1800.
  8. Solve for y: y = 1800 / (-3) = -600.
  9. Substitute y = -600 into x = -3y/2: x = -3*(-600)/2 = 1800/2 = 900.
  10. Check if x = 900 and y = -600 satisfy the original cost equation: 18*900 + 24*(-600) = 16200 - 14400 = 1800. Correct.
  11. Check revenue equation: 30*900 + 42*(-600) = 27000 - 25200 = 1800. Correct.
  12. The solution x = 900, y = -600 is mathematically valid but not feasible because y = -600 implies negative production of Brochure B, which is impossible. Therefore, no non-negative integer solution exists for break-even under these constraints. However, the question asks for the exact value of x that satisfies the system, regardless of feasibility. The algebraic solution gives x = 900.

Reference answer: 900

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