0580/42 May/Jun 2023, Question 9: Simultaneous Equations by Elimination

Part (ai) · 2 marks

Simplify the expression (3x²y⁴)³, giving the result in its simplest index form.

1. 13³ × (x²)³ × (y⁴)³
2. 2= 27x⁶y¹²

Answer27x⁶y¹²

Part (aii) · 3 marks

Simplify (16/(x¹⁶y⁸))^(−3/2), expressing the answer with positive indices.

1. 1(16/(x¹⁶y⁸))^(−3/2) = (x¹⁶y⁸/16)^(3/2)
2. 2√(x¹⁶y⁸/16) = x⁸y⁴/4
3. 3(x⁸y⁴/4)³ = x²⁴y¹²/64

Answerx²⁴y¹²/64

Part (bi) · 1 mark

Factorise the quadratic expression x² − 9.

1. 1x² − 9 = x² − 3² = (x + 3)(x − 3)

Answer(x + 3)(x − 3)

Part (bii) · 3 marks

Simplify the fraction with numerator x² − 9 and denominator 2xy − 6y + 5x − 15.

1. 12xy − 6y + 5x − 15 = 2y(x − 3) + 5(x − 3) = (x − 3)(2y + 5)
2. 2(x + 3)(x − 3)/((x − 3)(2y + 5))
3. 3= (x + 3)/(2y + 5)

Answer(x + 3)/(2y + 5)

Part (c) · 6 marks

Solve the simultaneous equations 2x + y = 7 and y = 5x² + 2x − 13, showing working and giving answers to 2 decimal places.

1. 1y = 7 − 2x, so 7 − 2x = 5x² + 2x − 13
2. 25x² + 4x − 20 = 0
3. 3x = (−4 ± √(4² − 4 × 5 × (−20)))/(2 × 5) = (−4 ± √416)/10
4. 4x = 1.639… or x = −2.439…
5. 5y = 7 − 2 × 1.639… = 3.720… and y = 7 − 2 × (−2.439…) = 11.879…
6. 6x = 1.64, y = 3.72 and x = −2.44, y = 11.88

Answerx = 1.64, y = 3.72 and x = −2.44, y = 11.88

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