AQA A-level Mathematics: Solving Simultaneous Equations by Elimination and Substitution (7357) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

This practice set covers statement B4 of Section B (Algebra and functions) in the AQA A-level Mathematics specification, first teaching 2017: solving simultaneous equations by elimination and substitution, including one linear and one quadratic equation. Read the study guide first if a method is unfamiliar, and use the revision notes for a quick recap. Course links: AQA A-level Mathematics hub, printable checklist and the free 10-minute diagnostics. Mark totals are shown in bold brackets. Show every step and give exact answers unless told otherwise.

Questions

1. Solve by substitution: y = 3 − 2x and 5x + 4y = 7. [3]

2. Solve by elimination: 7x − 2y = 9 and 3x + 5y = −4. [4]

3. Solve the simultaneous equations x − y = 2 and x² + y² = 52. [5]

4. Solve the simultaneous equations 2x + y = 4 and y = x² − 3x, giving exact answers in surd form. [5]

5. Show that the line y = 4x + 2 is a tangent to the curve y = x² + 2x + 3, and find the coordinates of the point of contact. [4]

6. A curve has equation y = x² + 5x + 7. A line has equation y = 3x + k, where k is a number.

(a) Show that the x-coordinates of any points of intersection satisfy x² + 2x + (7 − k) = 0. [1] (b) Determine k when the line just touches the curve. [2] (c) Determine all values of k for which the line and curve have no common point. [2] (d) Find the coordinates of the point of contact in the tangent case. [1]

7. Solve the simultaneous equations 3x + y = 14 and xy = 16. [5]

8. A rectangular plot has perimeter 40 m and area 84 m². Form a pair of simultaneous equations and find the length and width of the plot. [5]

9. A farm shop sells apples and pears by the kilogram. Four kilograms of apples and three kilograms of pears cost £19.60. Three kilograms of apples and five kilograms of pears cost £23.50. Find the price per kilogram of each fruit. [5]

10. The line y = x − 2 cuts the curve y = x² − 8x + 12 twice, at A and at B.

(a) Work out the coordinates of A and of B. [4] (b) Find the midpoint of AB and the exact length of AB. [3] (c) The line y = x + c is a tangent to the curve. Find c. [3]

11. The line y = x + c and the curve x² + 3y² = 12 are given.

(a) Show that the x-coordinates of any intersections satisfy 4x² + 6cx + 3c² − 12 = 0. [2] (b) Show that the discriminant of this equation is 192 − 12c². [2] (c) Find the values of c for which the line is a tangent to the curve. [1] (d) For the positive value of c, find the coordinates of the point of contact. [2]

Answers

1. Substitute: 5x + 4(3 − 2x) = 7 [1], so 5x + 12 − 8x = 7 and −3x = −5, giving x = 5/3 [1]. Then y = 3 − 10/3 = −1/3 [1]. x = 5/3, y = −1/3 [3] Examiner insight: Put the whole of 3 − 2x in brackets when you substitute; a missing bracket changes the sign of the 8x term and every later line then looks tidy but is wrong.

2. Multiply the first equation by 5 and the second by 2: 35x − 10y = 45 and 6x + 10y = −8 [1]. Add: 41x = 37 [1], so x = 37/41 [1]. Substitute: 3(37/41) + 5y = −4, so 5y = −275/41 and y = −55/41 [1]. x = 37/41, y = −55/41 [4] Examiner insight: An answer that is a fraction on both letters is a sign the arithmetic is probably right, not wrong; check it in the equation you did not use last rather than “fixing” it into integers.

3. From the first equation y = x − 2 [1]. Then x² + (x − 2)² = 52 gives 2x² − 4x − 48 = 0 [1], so x² − 2x − 24 = 0 and (x − 6)(x + 4) = 0 [1]. Hence x = 6 or x = −4 [1]. Using y = x − 2: (6, 4) and (−4, −6) [1] [5] Examiner insight: Pair the values using the linear equation. Substituting into x² + y² = 52 can give four apparent pairs, two of which are false.

4. Equate the two expressions for y: x² − 3x = 4 − 2x [1], so x² − x − 4 = 0 [1]. Quadratic formula: x = (1 ± √17)/2 [1]. Then y = 4 − 2x = 3 ∓ √17 [1]. ((1 + √17)/2, 3 − √17) and ((1 − √17)/2, 3 + √17) [1] [5] Examiner insight: When the question says “exact”, a decimal such as 2.56 loses the exactness. Keep the surd and match the signs carefully: the plus in x goes with the minus in y here.

5. Equate: x² + 2x + 3 = 4x + 2 [1], so x² − 2x + 1 = 0 [1]. This is (x − 1)² = 0, a repeated root (discriminant 4 − 4 = 0), so the line touches the curve once and is a tangent [1]. Then x = 1 and y = 4(1) + 2 = 6, so the point is (1, 6) [1]. [4] Examiner insight: A “show that” answer should state the conclusion in words, linking a repeated root to a tangent, as well as showing the algebra.

6. (a) x² + 5x + 7 = 3x + k rearranges to x² + 2x + (7 − k) = 0 [1]. (b) Discriminant: 2² − 4(7 − k) = 4k − 24 [1]. Tangent when 4k − 24 = 0, so k = 6 [1]. (c) No intersection when 4k − 24 < 0 [1], so k < 6 [1]. (d) With k = 6: x² + 2x + 1 = 0, so x = −1 and y = 3(−1) + 6 = 3. (−1, 3) [1] Examiner insight: “Does not meet” is a strict inequality on the discriminant, while “tangent” is equality. Writing k ≤ 6 includes the tangent case and so is not the answer to part (c).

7. From the first equation y = 14 − 3x [1]. Then x(14 − 3x) = 16 [1], giving 3x² − 14x + 16 = 0 [1]. Factorise: (3x − 8)(x − 2) = 0, so x = 8/3 or x = 2 [1]. Then y = 6 or y = 8, so (8/3, 6) and (2, 8) [1] [5] Examiner insight: Moving every term to one side before factorising matters here: with a negative leading coefficient it is easy to factorise a sign-flipped version and get the wrong roots.

8. Let the length be x and the width y. Perimeter: 2x + 2y = 40, so x + y = 20 [1]. Area: xy = 84 [1]. Substitute y = 20 − x: x² − 20x + 84 = 0 [1]. Factorise: (x − 6)(x − 14) = 0 [1]. The plot is 14 m by 6 m [1]. [5] Examiner insight: Both roots give the same rectangle with length and width swapped. State the dimensions once, with units, and say which is which.

9. Let apples cost £a per kg and pears £p per kg. Then 4a + 3p = 19.6 and 3a + 5p = 23.5 [1]. Multiply the first by 5 and the second by 3: 20a + 15p = 98 and 9a + 15p = 70.5 [1]. Subtract: 11a = 27.5 [1], so a = 2.5. Then 10 + 3p = 19.6, so p = 3.2 [1]. Check in the second equation: 7.5 + 16 = 23.5. Apples £2.50 per kg, pears £3.20 per kg [1] [5] Examiner insight: Define your letters with units at the start and answer in the context of the question. A bare “a = 2.5” leaves the final statement undone.

10. (a) Equate: x² − 8x + 12 = x − 2 [1], so x² − 9x + 14 = 0 [1]. This is (x − 2)(x − 7) = 0, so x = 2 or 7 [1]. With y = x − 2: A(2, 0) and B(7, 5) [1]. (b) Midpoint: (9/2, 5/2) [1]. Length: √(5² + 5²) = √50 [1] = 5√2 [1]. (c) Equate: x² − 9x + (12 − c) = 0 [1]. Discriminant: 81 − 4(12 − c) = 33 + 4c [1]. Tangent when 33 + 4c = 0, so c = −33/4 [1]. Examiner insight: In part (c) the constant term is 12 − c, not 12 + c. Moving c across the equals sign is the step where the sign slips.

11. (a) Substitute: x² + 3(x + c)² = 12 [1]. Expand: x² + 3x² + 6cx + 3c² = 12, so 4x² + 6cx + 3c² − 12 = 0 [1]. (b) Discriminant: (6c)² − 4(4)(3c² − 12) [1] = 36c² − 48c² + 192 = 192 − 12c² [1]. (c) Tangent when 192 − 12c² = 0, so c² = 16 and c = 4 or c = −4 [1]. (d) With c = 4: 4x² + 24x + 36 = 0, so x² + 6x + 9 = 0 and x = −3 [1]. Then y = −3 + 4 = 1. (−3, 1) [1] Examiner insight: A “show that” for the discriminant should display the substitution into b² − 4ac with the values of a, b and c written out before simplifying.

Where marks are usually lost

• Writing a quadratic in x but giving only x-values as the final answer.
• Forgetting that a tangent gives a repeated root, not two roots.
• Scaling one side of an equation only.
• Not putting a substituted expression in brackets before squaring.
• Losing a sign when moving a letter such as k or c across the equals sign.
• Using the quadratic equation, not the linear one, to find the partner value.
• Rounding surds to decimals when exact answers were requested.
• Not answering a context problem in context, with units.

Next steps

• Recap with the revision notes.
• Re-read the study guide for any method that felt shaky.
• Visit the AQA A-level Mathematics hub and tick items off the checklist.
• Take any of our free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

AQA A-level Mathematics (7357) specification, first teaching 2017, published by AQA. Statement B4, Section B: Algebra and functions.