This module covers pairs of equations where at least one is not a straight line, such as a line with a circle, or xy = 6 with x² + y² = 13. The method is always the same: use the simpler equation to eliminate one variable, solve the resulting quadratic, then find the matching partner value for every root. The same idea also turns word problems into equations and decides how many times a line meets a curve.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content and notation in your exam year, including any given formulae and calculator rules. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to expand brackets such as (x + 1)², factorise and solve quadratics, and recognise the equation of a circle. The discriminant matters from the third lesson onwards, so revise quadratic structure and discriminants first if b² − 4ac is not yet automatic. The circle equation itself is developed in circle coordinate methods.
An orienting example
Solve the simultaneous equations y = x + 1 and x² + y² = 25.
Step 1, eliminate y: substitute the line into the circle: x² + (x + 1)² = 25.
Step 2, expand and collect: x² + x² + 2x + 1 = 25, so 2x² + 2x − 24 = 0, which simplifies to x² + x − 12 = 0.
Step 3, solve: (x + 4)(x − 3) = 0, so x = −4 or x = 3.
Step 4, find each partner: use the line y = x + 1. When x = −4, y = −3. When x = 3, y = 4.
Answer: (−4, −3) and (3, 4).
Check: (−4)² + (−3)² = 16 + 9 = 25 and 3² + 4² = 9 + 16 = 25.
The line meets the circle at two points, and that matches the two roots of the quadratic. This link between roots and intersections is the thread through the whole module.
In which order should you study it?
- Substitute a line into a circle relationship: the core move, with care over brackets and pairing each x with its y.
- Eliminate a variable from a nonlinear pair: handles xy = k and squared pairs, where the quadratic is in x² and some roots must be rejected.
- Interpret a repeated solution geometrically: shows what a double root means for a line and a curve.
- Form a system from two constraints: turns a perimeter, a product or a diagonal into two equations.
- Check parameter values for a requested number of intersections: uses the discriminant to count intersections when the line contains an unknown constant.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Squaring a bracket wrongly, writing (x + 1)² as x² + 1 and losing the middle term.
- Stopping at the x-values, so the question asked for points but the answer has no y-coordinates.
- Pairing values the wrong way, putting a positive x with the y that belongs to the negative one.
- Keeping an impossible root, such as a y-value that would need x² to be negative.
- Solving the discriminant inequality carelessly, so that k² < 16 becomes k < 4 and the lower bound is lost.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt every question on paper, and finish each one by substituting your answers back into both original equations. That habit catches most slips before a teacher or an answer key does. Use the line and circle intersection explorer to confirm a result after you have worked it by hand, and the quadratic structure explorer to see how a discriminant changes the picture.
When you get something wrong, read the routing list at the end of the practice set and return to the lesson it names. Keep a record of the error types in the mistake log and retest queue, and retry a fresh question a few days later.
If progress stalls on the same habit, a teacher can look at your written solutions in online one-to-one Additional Mathematics tuition.