Higher June 2019 Paper 1 Q21
21 Solve the simultaneous equations
\[\begin{aligned} 2x + 3y &= 5p \\ y &= 2x + p \end{aligned}\]where \(p\) is a constant.
Give your answers in terms of \(p\) in their simplest form. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: substitution of \(2x + p\) for \(y\) | ||
| \(2x + 3(2x + p) = 5p\) | M1 | oe equation eg \(2x + 6x + 3p = 5p\) |
| \(6x + 2x = 5p - 3p\) or \(8x = 2p\) | M1dep | oe equation with terms collected condone incorrect expansion before rearrangement |
| Correct simplified terms \((x =)\ \dfrac{p}{4}\) or \(\dfrac{1}{4}p\) or \(0.25p\) and \((y =)\ \dfrac{3p}{2}\) or \(\dfrac{3}{2}p\) or \(1\dfrac{1}{2}p\) or \(1.5p\) | A2 | A1 one correct simplified term or otherwise correct terms for both with ‘\(p\)’ omitted eg \(x = 0.25\) and \(y = 1.5\) or correct unsimplified terms for both eg \(x = \dfrac{2p}{8}\) and \(y = \dfrac{6p}{4}\) |
| Alternative method 2: substitution of \(y - p\) for \(2x\) | ||
| \(y - p + 3y = 5p\) | M1 | oe equation |
| \(y + 3y = 5p + p\) or \(4y = 6p\) | M1dep | oe equation with terms collected |
| Correct simplified terms \((x =)\ \dfrac{p}{4}\) or \(\dfrac{1}{4}p\) or \(0.25p\) and \((y =)\ \dfrac{3p}{2}\) or \(\dfrac{3}{2}p\) or \(1\dfrac{1}{2}p\) or \(1.5p\) | A2 | A1 one correct simplified term or otherwise correct terms for both with ‘\(p\)’ omitted eg \(x = 0.25\) and \(y = 1.5\) or correct unsimplified terms for both eg \(x = \dfrac{2p}{8}\) and \(y = \dfrac{6p}{4}\) |
| Alternative method 3: elimination of \(x\) | ||
| \(y - 2x = p\) | M1 | oe with multiplication of both equations |
| \(y + 3y = 5p + p\) or \(4y = 6p\) | M1dep | oe addition must be seen if result is incorrect |
| Correct simplified terms \((x =)\ \dfrac{p}{4}\) or \(\dfrac{1}{4}p\) or \(0.25p\) and \((y =)\ \dfrac{3p}{2}\) or \(\dfrac{3}{2}p\) or \(1\dfrac{1}{2}p\) or \(1.5p\) | A2 | A1 one correct simplified term or otherwise correct terms for both with ‘\(p\)’ omitted eg \(x = 0.25\) and \(y = 1.5\) or correct unsimplified terms for both eg \(x = \dfrac{2p}{8}\) and \(y = \dfrac{6p}{4}\) |
| Alternative method 4: elimination of \(y\) | ||
| \(3y - 6x = 3p\) | M1 | oe with multiplication of both equations |
| \(2x - (-6x) = 5p - 3p\) or \(8x = 2p\) | M1dep | oe subtraction must be seen if result is incorrect |
| Correct simplified terms \((x =)\ \dfrac{p}{4}\) or \(\dfrac{1}{4}p\) or \(0.25p\) and \((y =)\ \dfrac{3p}{2}\) or \(\dfrac{3}{2}p\) or \(1\dfrac{1}{2}p\) or \(1.5p\) | A2 | A1 one correct simplified term or otherwise correct terms for both with ‘\(p\)’ omitted eg \(x = 0.25\) and \(y = 1.5\) or correct unsimplified terms for both eg \(x = \dfrac{2p}{8}\) and \(y = \dfrac{6p}{4}\) |