Revision mat · Simultaneous Equations (Linear & Quadratic)

No calculator11 marks

Question mix weighted by 94 real exam questions from 86 GCSE papers (2017–24), so the most common question types get the most space. Show your working.

Substitution $y=mx+c \;\Rightarrow\; \text{replace } y$Form a quadratic $ax^2+bx+c=0$Line & circle $x^2+y^2=r^2$Discriminant test $b^2-4ac$
Line & quadratic≈11% of real exam Qs
1The graph shows the curve $y = x^2$ (red) and the line $y = x + 6$ (blue).
They meet at two points.
Use the graph to write down the coordinates of both points of intersection.
-4-224-2246810xy
Answer: (3 marks)
2The graph shows the curve $y = x^2$ (red) and the line $y = -2x + 3$ (blue).
They meet at two points.
Use the graph to write down the coordinates of both points of intersection.
-4-224-2246810xy
Answer: (3 marks)
★ Exam capstonemixed & other forms ≈89% of real exam Qs
3Solve the simultaneous equations $$\begin{aligned} x^2 + y^2 &= 25 \\ y &= 3x - 15 \end{aligned}$$
Answer: (5 marks)

Mark scheme · Simultaneous Equations (Linear & Quadratic)

Total: 11 marks

Award the marks shown for each correct step, then add up the total out of 11. A method mark counts even if the final answer is wrong.

1The graph shows the curve $y = x^2$ (red) and the line $y = x + 6$ (blue).
They meet at two points.
Use the graph to write down the coordinates of both points of intersection.[3 marks]
Method
Read the two crossing points off the graph.
The line meets the curve at $(-2, 4)$ and $(3, 9)$.
Check: substituting each $x$ into $y = x^2$ gives the matching $y$-value.
Answer: $(-2, 4) and (3, 9)$
Marks
✔1 markRead off the first intersection point
✔1 markRead off the second intersection point
✔1 markBoth coordinates correct: (-2, 4) and (3, 9)
2The graph shows the curve $y = x^2$ (red) and the line $y = -2x + 3$ (blue).
They meet at two points.
Use the graph to write down the coordinates of both points of intersection.[3 marks]
Method
Read the two crossing points off the graph.
The line meets the curve at $(-3, 9)$ and $(1, 1)$.
Check: substituting each $x$ into $y = x^2$ gives the matching $y$-value.
Answer: $(-3, 9) and (1, 1)$
Marks
✔1 markRead off the first intersection point
✔1 markRead off the second intersection point
✔1 markBoth coordinates correct: (-3, 9) and (1, 1)
3Solve the simultaneous equations $$\begin{aligned} x^2 + y^2 &= 25 \\ y &= 3x - 15 \end{aligned}$$[5 marks]
Method
Substitute $y = 3x - 15$ into $x^2 + y^2 = 25$.
This gives a quadratic in $x$; solving it gives $x = 4$ or $x = 5$.
Find $y$ from the line: when $x = 4$, $y = -3$; when $x = 5$, $y = 0$.
Solutions: $(4, -3)$ and $(5, 0)$.
Answer: $(4, -3) and (5, 0)$
Marks
✔1 markSubstitute the line into the circle
✔1 markForm a quadratic in x
✔1 markSolve for x: 4 and 5
✔1 markFind the matching y-values
✔1 mark(4, -3) and (5, 0)
mathedup.co.uk · sheet RHZL