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Exploring linear functions

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A sports equipment supplier allocates tennis rackets and cricket bats to schools in Brisbane. The number of tennis rackets allocated is modelled by the linear function y = 42x + 46, where x is the number of schools and y is the total number of rackets.

The number of cricket bats is allocated such that each school receives exactly 46 bats, and the total number of bats is 46 more than twice the number of rackets allocated. What is the exact number of schools (x) that received equipment?

Show your working using substitution and solving the linear equation.

Concept illustrations

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A straight-line relationship. Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.-2-2-1-111223344xy
Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.
Keep both sides balanced. The level beam represents equality, not measured weights. Subtract 6 from both sides of x + 6 = 14 to leave x = 8.x + 614−6−6x = 8
The level beam represents equality, not measured weights. Subtract 6 from both sides of x + 6 = 14 to leave x = 8.

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Explanation

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Worked solution

  1. The number of tennis rackets allocated is given by the linear function y = 42x + 46, where x is the number of schools and y is the total number of rackets.
  2. Each school receives 46 cricket bats, so the total number of cricket bats is 46x.
  3. The total number of cricket bats is also described as 46 more than twice the number of rackets allocated. So, total bats = 2y + 46.
  4. Substitute the expression for y from step 1 into the equation from step 3: 46x = 2(42x + 46) + 46.
  5. Expand the right-hand side: 46x = 84x + 92 + 46.
  6. Simplify the constants: 46x = 84x + 138.
  7. Subtract 84x from both sides: 46x - 84x = 138.
  8. Simplify: -38x = 138.
  9. Divide both sides by -38: x = 138 / (-38).
  10. Simplify the fraction: divide numerator and denominator by 2 to get x = -69 / 19.
  11. The result is a negative number, but the number of schools cannot be negative. This indicates a contradiction in the setup or interpretation.
  12. Re-examining the problem: the statement says 'the total number of bats is 46 more than twice the number of rackets'. This means 46x = 2y + 46. Substituting y = 42x + 46 gives 46x = 2(42x + 46) + 46 = 84x + 92 + 46 = 84x + 138. Rearranging: 46x - 84x = 138 → -38x = 138 → x = -138/38 = -69/19 ≈ -3.63. This is invalid as x must be a positive integer. Therefore, the initial interpretation must be incorrect.

Reference answer: The number of schools cannot be determined as the equation yields a negative value, which is not possible for a count of schools.

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