Simultaneous Equations
Simultaneous equations in SPM Additional Mathematics almost always pair one linear equation with one non-linear equation such as a circle, parabola or hyperbola. The exam tests substitution, the link between the number of solutions and the number of intersection points, and the use of the discriminant to find an unknown constant.
Method
Make one variable the subject of the linear equation, substitute into the non-linear equation, and solve the resulting quadratic. Back-substitute to find the paired value. Geometrically, each real solution is an intersection point; a repeated root means the line is a tangent to the curve, and no real root means they do not meet.
Worked example
Solve y = 2x + 1 and x² + y² = 13. Substitute: x² + (2x+1)² = 13, so x² + 4x² + 4x + 1 = 13, giving 5x² + 4x − 12 = 0. Factorise: (5x − 6)(x + 2) = 0, so x = 6/5 or x = −2. Then y = 2(6/5) + 1 = 17/5, or y = 2(−2) + 1 = −3. The solutions are (6/5, 17/5) and (−2, −3). Substituting both pairs into x² + y² = 13 verifies them.
Traps
Reporting only the x-values; pairing an x-value with the wrong y-value; squaring an equation and accepting extraneous roots without checking; expansion errors in (2x+1)².