NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Simplifying linear expressions

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Required skills (click to reveal)

A sports equipment supplier in Sydney is packing tennis rackets and tennis balls into boxes. Each box contains 39 tennis rackets and 92 tennis balls.

The total number of items in x boxes is given by the expression 39x + 92. If the supplier uses 3x boxes for rackets and 2x boxes for balls, and each box holds the same number of items, what is the simplified form of the total number of items across all boxes?

Show all steps using the distributive, associative, and commutative properties. Provide the exact simplified linear expression as a single algebraic expression with integer coefficients and constant.

Concept illustrations

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Combining like terms as equal tiles. Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.xxxxx3x2x3x + 2x = 5x
Each tile represents x. Three x-tiles and two x-tiles make five x-tiles. A constant tile represents a different quantity.
Distributive law as a rectangle. A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.x433x123(x + 4) = 3x + 12
A rectangle of height 3 and total width x + 4 splits into areas 3x and 12. Therefore 3(x + 4) = 3x + 12. Algebraic lengths are schematic.

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Formula and explanation (click to reveal)

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The total number of items in x boxes is given as 39x + 92. This expression represents the sum of tennis rackets and tennis balls across x boxes.
  2. The supplier uses 3x boxes for rackets and 2x boxes for balls. Since each box holds the same number of items, the total number of items is the sum of items in all boxes.
  3. Each box contains 39 rackets and 92 balls, so the total items from 3x boxes for rackets is 39 * 3x = 117x.
  4. The total items from 2x boxes for balls is 92 * 2x = 184x.
  5. The total number of items across all boxes is 117x + 184x. This uses the commutative property: addition order does not affect sum.
  6. Apply the associative property: group like terms: (117x + 184x) = (117 + 184)x.
  7. Add coefficients: 117 + 184 = 301. So, 301x.
  8. The expression 301x is a linear expression with integer coefficient and no constant term.
  9. The distributive property was used in step 3 and 4: a(bx) = (ab)x, e.g., 39 * 3x = (39*3)x = 117x.
  10. The commutative and associative properties were used to reorder and group terms: 117x + 184x = (117 + 184)x.
  11. The simplified linear expression is 301x. This matches the form of a linear expression with integer coefficients and no constant.
  12. Check: 3x boxes at 39 rackets each = 117x rackets; 2x boxes at 92 balls each = 184x balls; total = 117x + 184x = 301x. Correct.

Reference answer: 301x

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