Higher June 2023 Paper 1 Q16
16 Solve the simultaneous equations
\(2x - 5y = 13\)
\(3x + 4y = 8\)
[4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – equates coefficients and eliminates an unknown | ||
| \(8x - 20y = 52\) and \(15x + 20y = 40\) or \(6x - 15y = 39\) and \(6x + 8y = 16\) | M1 | oe equates coefficients of one unknown allow one term error |
| \(8x + 15x = 52 + 40\) or \(23x = 92\) or \(-15y - 8y = 39 - 16\) or \(-23y = 23\) | M1dep | oe eliminates an unknown must be correct for their equations |
| \(x = 4\) and \(y = -1\) | A2 | A1 \(x = 4\) from correct method or \(y = -1\) from correct method |
| Alternative method 2 – substitutes for \(x\) | ||
| \(x = 6.5 + 2.5y\) or \(x = \dfrac{8}{3} - \dfrac{4}{3}y\) | M1 | oe makes \(x\) the subject of one equation allow one term error |
| \(3(6.5 + 2.5y) + 4y = 8\) or \(11.5y = -11.5\) or \(2\left(\dfrac{8}{3} - \dfrac{4}{3}y\right) - 5y = 13\) or \(-\dfrac{23}{3}y = \dfrac{23}{3}\) | M1dep | oe eliminates \(x\) must be correct for their rearrangement |
| \(x = 4\) and \(y = -1\) | A2 | A1 \(y = -1\) from this method |
| Alternative method 3 – substitutes for \(y\) | ||
| \(y = 0.4x - 2.6\) or \(y = 2 - 0.75x\) | M1 | oe makes \(y\) the subject of one equation allow one term error |
| \(3x + 4(0.4x - 2.6) = 8\) or \(4.6x = 18.4\) or \(2x - 5(2 - 0.75x) = 13\) or \(5.75x = 23\) | M1dep | oe eliminates \(y\) must be correct for their rearrangement |
| \(x = 4\) and \(y = -1\) | A2 | A1 \(x = 4\) from this method |
| Alternative method 4 – makes the same unknown the subject in both equations | ||
| \(x = 6.5 + 2.5y\) or \(x = \dfrac{8}{3} - \dfrac{4}{3}y\) or \(y = 0.4x - 2.6\) or \(y = 2 - 0.75x\) | M1 | oe makes \(y\) or \(x\) the subject of one equation allow one term error |
| \(6.5 + 2.5y = \dfrac{8}{3} - \dfrac{4}{3}y\) or \(\dfrac{23}{6}y = -\dfrac{23}{6}\) or \(0.4x - 2.6 = 2 - 0.75x\) or \(1.15x = 4.6\) | M1dep | oe makes \(y\) or \(x\) the subject of both equations (maximum one term error) and eliminates \(y\) or \(x\) must be correct for their rearrangements |
| \(x = 4\) and \(y = -1\) | A2 | A1 \(x = 4\) from correct method or \(y = -1\) from correct method |
Additional guidance
| Up to M2 may be awarded for correct work seen in multiple attempts, even if not subsequently used | |
| In alts 2, 3 and 4 allow rounding or truncating to 1dp or better for up to M1M1 eg (Alt 4) \(6.5 + 2.5y = 2.7 - 1.3y\) | M1M1 |
| Answers from trial and improvement or with no working score 0 or 4 |