Higher June 2025 Paper 3 Q21
21 Solve algebraically the simultaneous equations (5)
\[\begin{aligned} 3x^2 + 2y^2 &= 44 \\ 3x + y &= 2 \end{aligned}\]| Answer | Mark | Mark scheme |
|---|---|---|
| \(x = -\dfrac{6}{7}\), \(y = \dfrac{32}{7}\) \(x = 2\), \(y = -4\) | M1 | for substitution of a rearranged equation into a correct equation to form an equation in one variable. eg \(3x^2 + 2(2 - 3x)^2 = 44\) or \(3\left(\dfrac{2 - y}{3}\right)^2 + 2y^2 = 44\) |
| M1 | (dep on first M1) for multiplying out all brackets and collecting terms to form a simplified three term quadratic in any form of \(ax^2 + bx + c\ (= 0)\) where at least 2 coefficients (\(a\), \(b\), \(c\)) are correct eg \(21x^2 - 24x - 36\ (= 0)\) or \(7x^2 - 8x - 12\ (= 0)\) or \(7y^2 - 4y - 128\ (= 0)\) | |
| M1 | (dep on first M1) for a suitable method to solve their 3 term quadratic using any correct method, for factorising, eg \((7x + 6)(x - 2)\) or \((7x + 6)(3x - 6)\) or \((21x + 18)(x - 2)\) or \((7y - 32)(y + 4)\) or correct use of formula, eg \(\dfrac{8 \pm \sqrt{(-8)^2 - 4 \times 7 \times -12}}{2 \times 7}\) or \(\dfrac{4 \pm \sqrt{(-4)^2 - 4 \times 7 \times -128}}{2 \times 7}\) or completing the square | |
| M1 | (dep on first M1) for substituting their 2 found values of \(x\) or \(y\) in a suitable equation or (dep on first M1) for one correct pair of values following from a correct quadratic | |
| A1 | for \(x = -\dfrac{6}{7}\) oe, \(y = \dfrac{32}{7}\) oe and \(x = 2\), \(y = -4\) |
Additional guidance
Allow \((\pm 2 \pm 3x)\) for \((2 - 3x)\)
(or \(\left(\dfrac{\pm 2 \pm y}{3}\right)\) for \(\left(\dfrac{2 - y}{3}\right)\))
Implied by a correct equation (simplified or unsimplified) in terms of \(x\) or \(y\)
eg \(3x^2 + 2(4 - 12x + 9x^2) = 44\)
or \(3x^2 + 8 - 24x + 18x^2 = 44\)
or \(21x^2 - 24x = 36\)
Look out for signs reversed
The quadratic does not have to equal 0, ie accept \(21x^2 - 24x = 36\)
Can be implied by both \(x\) values or both \(y\) values correct (condone incorrect labelling) if the quadratic is correct
If using the quadratic formula (condone one sign error, omission of brackets around the \(b\) in the \(b^2 - 4ac\) and the fraction line not being under the \(-\) in the \(-b\). Allow some simplification – as far as eg \(\dfrac{8 \pm \sqrt{64 + 336}}{14}\)
or if factorising allow brackets which expand to give 2 out of 3 terms correct for their quadratic
Condone substitution into their \((\pm 2 \pm 3x)\) or \(\left(\dfrac{\pm 2 \pm y}{3}\right)\)
Allow \(-0.85(7\ldots)\) or \(-0.86\) for \(-\dfrac{6}{7}\)
Allow 4.57(1…) for \(\dfrac{32}{7}\)
If values of \(x\) or \(y\) are incorrect then working must be shown
Accept as coordinates
Assume correct pairing unless clearly incorrect eg \(\left(-\tfrac{6}{7}, -4\right), \left(2, \tfrac{32}{7}\right)\)
Allow \(-0.85(7\ldots)\) or \(-0.86\) for \(-\dfrac{6}{7}\)
Allow 4.57(1…) for \(\dfrac{32}{7}\)
If an answer is shown in the range in working and then incorrectly rounded award full marks
A correct answer with no supportive working gets 0 marks