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

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A library in Melbourne is cataloguing books using a linear system where the total number of books, B, is related to the number of shelves, s, by the equation 22s + 26 = 3B. If the library uses exactly 15 shelves, what is the exact number of books catalogued?

Solve algebraically and verify by substitution. Provide the exact integer or fraction as the final answer.

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

Worked solution and answer (click to reveal)

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Worked solution

  1. The equation relating the number of books B to the number of shelves s is given as 22s + 26 = 3B.
  2. The library uses exactly 15 shelves, so substitute s = 15 into the equation.
  3. Substitute s = 15: 22(15) + 26 = 3B.
  4. Calculate 22 × 15: 22 × 15 = 330.
  5. Add 26 to 330: 330 + 26 = 356.
  6. Now the equation is 356 = 3B.
  7. Solve for B by dividing both sides by 3: B = 356 / 3.
  8. Perform the division: 356 ÷ 3 = 118.666..., which is 118 and 2/3.
  9. Since the number of books must be a whole number, and 356 is not divisible by 3, the result is a fraction: 356/3.
  10. Verify by substitution: substitute B = 356/3 into 3B: 3 × (356/3) = 356.
  11. Check the left side: 22(15) + 26 = 330 + 26 = 356. Both sides are equal, so the solution is correct.
  12. The exact number of books catalogued is 356/3, which is 118 and 2/3. Since books must be whole, and the equation yields a fractional result, the library must have 118 full books and 2/3 of a book, which is not possible. However, the question asks for the exact number as per the equation, so the answer is 356/3.

Reference answer: 356/3

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