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)

In a community garden in Melbourne, a rectangular planting bed has a length represented by the expression (77x - 3) metres and a width represented by (73x + 5) metres. The perimeter is calculated using the formula P = 2(length + width).

After expanding and simplifying the expression for the perimeter using the distributive, associative, and commutative properties, what is the exact simplified form of the perimeter as a single linear algebraic expression with integer coefficients and constant? Show all steps and provide the final answer as 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

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The length of the rectangular planting bed is (77x - 3) m and the width is (73x + 5) m. The perimeter P of a rectangle is given by P = 2(length + width).
  2. Substitute the expressions for length and width into the perimeter formula: P = 2[(77x - 3) + (73x + 5)].
  3. Apply the commutative and associative properties to group like terms inside the parentheses: (77x + 73x) + (-3 + 5).
  4. Simplify the grouped terms: 77x + 73x = 150x, and -3 + 5 = 2. So, the sum is 150x + 2.
  5. Now substitute back into the perimeter formula: P = 2(150x + 2).
  6. Apply the distributive property: 2 × 150x = 300x, and 2 × 2 = 4.
  7. Combine the results: P = 300x + 4.
  8. The expression 300x + 4 is a linear algebraic expression with integer coefficients and a constant term.
  9. Verify the simplification using the distributive property: 2(150x + 2) = 2×150x + 2×2 = 300x + 4.
  10. Check that all terms are simplified and no further combining is possible. The expression is in standard form: ax + b.
  11. The simplified perimeter is 300x + 4 metres.
  12. Final answer confirmed: the perimeter is exactly 300x + 4 metres.

Reference answer: 300x + 4 m

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