2.6 Simultaneous linear equations
- Syllabus
- 2017
- Topic
- 2.6
- Level
- Higher
Learning objectives
2.6A Exact solutions of simultaneous linear equationsCalculate exact solutions of two simultaneous linear equations in two unknowns.2.6HA Higher-tier simultaneous linear equationsCalculate exact solutions of higher-tier simultaneous linear equations in two unknowns.2.6HB Graphical interpretation of simultaneous equationsinterpret the equations as lines and the common solution as the point of intersection
Solve simultaneous linear equations exactly
A simultaneous solution is one ordered pair that satisfies both linear equations at the same time. Elimination and substitution reduce the pair to one equation in one unknown.
| Structure | Efficient method |
|---|---|
| one variable already isolated | substitute it into the other equation |
| equal or opposite coefficients | add or subtract to eliminate directly |
| coefficients have a small common multiple | scale one or both equations, then eliminate |
For 7x−2y=34 and 3x+5y=−3, multiply the first by 5 and the second by 2: 35x−10y=170 and 6x+10y=−6. Adding gives 41x=164, so x=4 and then y=−3.
Keep fractions exact rather than rounding intermediate results. After finding one variable, substitute into an original equation to find the other.
Scaling an equation means multiplying every term on both sides. Eliminating one variable is only the first half: the final answer must give and verify both values.
Control higher-tier simultaneous linear equations
Higher-tier systems use the same valid elimination or substitution principles, but may require clearing fractions, managing decimals, or choosing multipliers that avoid unnecessary complexity.
| Step | Control decision |
|---|---|
| standardise | clear denominators and write both equations as ax+by=c |
| choose | eliminate the variable needing the simplest integer multipliers |
| combine | add or subtract whole equations with signs visible |
| recover | substitute the exact first value to find the second |
| verify | test the ordered pair in both original equations |
For (x+2y)/3=5 and 2x−y/2=4, first write x+2y=15 and 4x−y=8. This exposes integer coefficients before elimination.
If elimination produces a false statement such as 0=5, the lines are parallel and there is no solution. If it produces 0=0, the equations describe the same line and have infinitely many solutions.
Do not convert exact fractions to rounded decimals mid-solution. Approximation can make a correct common solution fail one of the original equations.
Interpret simultaneous equations as intersecting lines
Each linear equation in two unknowns represents a straight line. A point satisfying both equations lies on both lines, so the simultaneous solution is their point of intersection.
| Stage | Graphical action |
|---|---|
| rearrange | write each equation in a plottable form such as y=mx+c |
| plot | use two or more accurate points for each line |
| intersect | read the common coordinate using the graph scale |
| report | write x from the horizontal coordinate and y from the vertical |
| check | substitute the read values into both equations |
For y−x−2=0 and 2y+x=1, the lines are y=x+2 and y=(1−x)/2. They intersect at (−1,1), so x=−1 and y=1.
Intersecting lines give one solution; distinct parallel lines give none; coincident lines give infinitely many. A graph may give only an approximate coordinate unless the intersection is exactly readable.
A point on only one line is not a simultaneous solution. Drawing must cover the actual intersection and use a scale precise enough for the requested accuracy.