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

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A café in Perth offers two types of specialty coffee: a flat white costs $4.50 to make and sells for $7.20, while a latte costs $5.10 to make and sells for $8.40. The café has a daily budget of $360 for ingredients and can prepare at most 60 drinks per day.

Let x be the number of flat whites and y be the number of lattes. The café wants to know how many flat whites (x) must be sold to exactly break even on total cost and total revenue, assuming all drinks are sold.

Solve the system of equations 4.50x + 5.10y = 360 and 7.20x + 8.40y = 360 exactly and give the exact integer value of x.

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Worked solution

  1. The café has a daily ingredient budget of $360, so the cost equation is 4.50x + 5.10y = 360, where x is the number of flat whites and y is the number of lattes.
  2. Total revenue from selling x flat whites and y lattes is 7.20x + 8.40y. For break-even, revenue must equal cost, so 7.20x + 8.40y = 360.
  3. We now solve the system of equations: 4.50x + 5.10y = 360 and 7.20x + 8.40y = 360.
  4. Multiply the first equation by 8 and the second by 5 to eliminate decimals and align coefficients for elimination: 8*(4.50x + 5.10y) = 8*360 → 36.00x + 40.80y = 2880; 5*(7.20x + 8.40y) = 5*360 → 36.00x + 42.00y = 1800.
  5. Subtract the first modified equation from the second: (36.00x + 42.00y) - (36.00x + 40.80y) = 1800 - 2880 → 1.20y = -1080.
  6. Solve for y: y = -1080 / 1.20 = -900.
  7. Substitute y = -900 into the first equation: 4.50x + 5.10*(-900) = 360 → 4.50x - 4590 = 360.
  8. Solve for x: 4.50x = 360 + 4590 = 4950 → x = 4950 / 4.50.
  9. Calculate x: 4950 ÷ 4.50 = 495000 ÷ 450 = 1100 / 1 = 1100.
  10. The solution x = 1100, y = -900 is not feasible because y is negative, meaning the café cannot produce a negative number of lattes.
  11. Re-evaluate the break-even condition: the question asks for the number of flat whites (x) that must be sold to break even, assuming all drinks are sold and the total cost is $360. But the revenue equation 7.20x + 8.40y = 360 implies that total revenue equals $360, which is the same as the cost budget. This means the café must sell drinks worth exactly $360 in revenue to break even on a $360 cost budget.
  12. Since the solution yields a negative number of lattes, it is not physically possible. Therefore, no combination of x and y with non-negative integers satisfies both cost and revenue equal to $360. The break-even point cannot be achieved under these constraints. However, the question asks for the exact integer value of x that satisfies the system, so we return the mathematical solution even if it is not feasible.

Reference answer: 1100

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