One Linear + One Non-Linear (Quadratic)

The 0606 version of simultaneous equations pairs a line with a curve. There is one method, substitute from the linear, and a handful of habits that decide whether the routine scores fully.

The routine in full

Solve $x + 2y = 7$ and $x^2 + y^2 = 10$.

1. Rearrange the linear: $x = 7 - 2y$ (never try to rearrange the quadratic)
2. Substitute with brackets intact: $(7 - 2y)^2 + y^2 = 10$ ← this line is the M mark
3. Expand and collect: $49 - 28y + 4y^2 + y^2 = 10 \to 5y^2 - 28y + 39 = 0$
4. Solve: $(5y - 13)(y - 3) = 0 \to y = \tfrac{13}{5}$ or $y = 3$
5. Back-substitute into the linear: $y = 3 \to x = 1$; $y = \tfrac{13}{5} \to x = 7 - \tfrac{26}{5} = \tfrac{9}{5}$
6. Present as pairs: $(1, 3)$ and $\left(\tfrac{9}{5}, \tfrac{13}{5}\right)$

Choose which variable to isolate by friction: here $x = 7 - 2y$ avoids fractions, while $y = \dfrac{7 - x}{2}$ invites them. On the non-calculator paper that choice is the difference between clean working and a fraction swamp.

The three habits that protect the marks

Brackets. $(7 - 2y)^2$ written before expansion, squaring a two-term expression mentally is the topic’s most common crash site.

Back-substitute into the linear. The quadratic accepts wrong pairings happily (it’s symmetric in $\pm$); the linear doesn’t. Less algebra, built-in validation.

Pair the answers. Each $y$ with its own $x$, as coordinates. Lists of $x$-values and $y$-values without pairing can drop the final mark even when every number is right (the pairing discipline).

When not to solve at all

If the question asks only whether (or for what $k$) solutions exist, stop at step 3 and apply the discriminant, solving fully answers a question that wasn’t asked and spends time the paper doesn’t give.

Common mistakes

• Substitution attempted from the quadratic
• Brackets dropped at the substitution step
• The fraction-heavy variable isolated when the clean one was available
• Back-substitution into the quadratic, generating phantom pairings
• Solutions unpaired in the final answer

Full topic context: Simultaneous Equations notes · the geometry: points of intersection.