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 Perth is ordering classroom supplies. The order includes 16 boxes of markers and 60 boxes of highlighters.

Each box of markers contains x markers, and each box of highlighters contains 3x - 4 highlighters. The total number of writing items ordered is given by the expression 16x + 60(3x - 4).

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

Final answer must be in the form of a simplified expression.

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 expression given is 16x + 60(3x - 4). This represents the total number of writing items ordered, with 16 boxes of markers (each containing x markers) and 60 boxes of highlighters (each containing 3x - 4 highlighters).
  2. Apply the distributive property to expand 60(3x - 4). Multiply 60 by each term inside the parentheses: 60 * 3x = 180x and 60 * (-4) = -240.
  3. Substitute the expanded terms back into the expression: 16x + 180x - 240.
  4. Use the commutative property of addition to rearrange the terms: 16x + 180x - 240 becomes 16x + 180x - 240 (order does not affect sum).
  5. Use the associative property to group like terms: (16x + 180x) - 240.
  6. Combine the like terms: 16x + 180x = 196x.
  7. The simplified expression is now 196x - 240.
  8. Verify the simplification by checking the coefficients and constant: 16x + 60(3x - 4) = 16x + 180x - 240 = 196x - 240.
  9. Confirm that all coefficients and constants are integers and the expression is linear.
  10. The expression is fully simplified and in standard form: 196x - 240.
  11. Check by substituting a value for x, for example x = 1: 16(1) + 60(3(1) - 4) = 16 + 60(-1) = 16 - 60 = -44. 196(1) - 240 = -44. Match confirms correctness.
  12. The simplified form of the expression is 196x - 240.

Reference answer: 196x - 240

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