Simultaneous Equations (Copy)
SIMULTANEOUS EQUATIONS — O LEVEL & IGCSE ADDITIONAL MATHEMATICS FORMULA SHEET + COMMON MISTAKES
| Topic | Formula / Rule | Common Mistake |
|---|---|---|
| Simultaneous Equations | Solve equations together | Solving separately |
| Linear Simultaneous Equations | Two linear equations solved together | Forgetting both equations must be satisfied |
| Linear & Quadratic System | Substitute line into curve | Substituting wrong equation |
| Substitution Method | Replace one variable using another equation | Partial substitution |
| Elimination Method | Eliminate one variable | Wrong multiplication |
| Graphical Method | Intersection points are solutions | Reading graph inaccurately |
| Intersection Point | Same x and y satisfy both equations | Giving only x-values |
| Linear Equation | y = mx + c | Mixing m and c |
| Quadratic Equation | y = ax² + bx + c | Forgetting a ≠0 |
| Circle Equation | x² + y² = r² | Forgetting squares |
| Substitute into Quadratic | Form quadratic equation | Rearranging incorrectly |
| Quadratic Formula | x = (−b ± √(b² − 4ac))/2a | Missing brackets |
| Discriminant | Δ = b² − 4ac | Forgetting square on b |
| Nature of Solutions | Δ > 0 → 2 intersections | Mixing conditions |
| Nature of Solutions | Δ = 0 → tangent | Thinking two intersections exist |
| Nature of Solutions | Δ < 0 → no real intersections | Trying to force answers |
| Tangent Condition | Line touches curve once | Drawing crossing graph |
| Solving for y | Substitute x-values back | Forgetting y-values |
| Exact Coordinates | Write full coordinate pair | Writing only one coordinate |
| Rearranging Equations | Make variable subject carefully | Sign errors |
| Simultaneous with Circle | Substitute line into circle | Expanding incorrectly |
| Simultaneous with Cubic | May have up to 3 solutions | Assuming only 2 |
| Simultaneous with Quadratic | May have 0, 1 or 2 solutions | Assuming always 2 |
| Graph Interpretation | Solutions = intersections | Mixing turning points with roots |
| Elimination | Match coefficients first | Adding wrong equations |
| Fractional Equations | Multiply through by denominator first | Leaving fractions |
| Non-linear System | Usually forms quadratic | Incorrect degree formed |
| Repeated Root | One repeated solution | Forgetting tangent meaning |
| Coordinates on Graph | (x, y) | Reversing order |
| Verification | Substitute into BOTH equations | Checking one equation only |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
QUICK METHODS TABLE
| Method | Main Idea |
|---|---|
| Substitution | Replace one variable |
| Elimination | Remove one variable |
| Graphical | Find intersections visually |
| Quadratic Substitution | Form quadratic after substitution |
QUICK NATURE OF SOLUTIONS TABLE
| Discriminant | Meaning |
|---|---|
| Δ > 0 | Two distinct intersections |
| Δ = 0 | One repeated intersection |
| Δ < 0 | No real intersections |
QUICK GRAPH RELATIONSHIPS
| Graphs | Possible Solutions |
|---|---|
| Line and line | 0 or 1 solution |
| Line and quadratic | 0, 1 or 2 solutions |
| Line and circle | 0, 1 or 2 solutions |
| Line and cubic | Up to 3 solutions |
| Quadratic and quadratic | Up to 4 solutions |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
COMMON EQUATION FORMS
| Type | Equation |
|---|---|
| Straight Line | y = mx + c |
| Quadratic | y = ax² + bx + c |
| Circle | x² + y² = r² |
| Cubic | y = ax³ + bx² + cx + d |
QUICK SIMULTANEOUS STEPS
| Step | Action |
|---|---|
| 1 | Rearrange equations |
| 2 | Substitute or eliminate |
| 3 | Form polynomial |
| 4 | Solve for x |
| 5 | Find y-values |
| 6 | Write coordinates |
EXAM HACKS
| Situation | Fast Trick |
|---|---|
| Line with curve | Substitute line into curve |
| Tangent question | Check if Δ = 0 |
| Elimination | Match coefficients first |
| Graph questions | Intersections = solutions |
| Circle equations | Expand brackets carefully |
| Exact answers | Leave surds exact |
| Multiple roots | Write all coordinates |
| Verification | Check both equations |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
MOST COMMON LOSSES OF MARKS
| Mistake | Why Marks Are Lost |
|---|---|
| Wrong substitution | Entire solution incorrect |
| Forgetting y-values | Incomplete coordinates |
| Arithmetic sign errors | Wrong roots |
| Incorrect expansion | Wrong quadratic formed |
| Wrong discriminant use | Wrong number of solutions |
| Writing only x-values | Coordinate marks lost |
| Not checking both equations | Invalid solutions |
| Graph reading errors | Incorrect coordinates |
| Forgetting tangent condition | Wrong interpretation |
| Poor rearrangement | Algebra collapses early |
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.