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 school in Brisbane is ordering classroom supplies. The order includes 20 boxes of pencils and 37 boxes of erasers.

Each box of pencils contains x pencils, and each box of erasers contains 2x erasers. The total number of writing items ordered is given by the expression 20x + 37(2x).

Simplify this linear expression using the distributive, associative, and commutative properties. Provide the exact simplified form 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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The expression given is 20x + 37(2x). This represents the total number of writing items, with 20 boxes of pencils (each containing x pencils) and 37 boxes of erasers (each containing 2x erasers).
  2. Apply the distributive property: 37(2x) = 37 × 2 × x = 74x.
  3. Substitute the result back into the expression: 20x + 74x.
  4. Use the commutative property of addition to reorder terms: 20x + 74x = 74x + 20x.
  5. Use the associative property to group like terms: (20x + 74x).
  6. Add the coefficients: 20 + 74 = 94.
  7. Combine the terms: 94x.
  8. The simplified expression is 94x, which is a linear expression with integer coefficient and no constant term.
  9. Verify using substitution: let x = 1. Then original expression: 20(1) + 37(2×1) = 20 + 74 = 94. Simplified: 94(1) = 94. Match.
  10. Check algebraic structure: 20x + 74x = (20 + 74)x = 94x. All steps follow the distributive, commutative, and associative properties.
  11. The simplified form is 94x, with integer coefficient and no constant. This is the final answer.
  12. The total number of writing items simplifies to 94x, which is a single algebraic expression with integer coefficients and no constant.

Reference answer: 94x

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