Form 4 Additional Mathematics Bab 3: Systems of Equations
3.1 Systems of Linear Equations in Three Variables
General form of a linear equation in three variables: ax + by + cz = d. A system needs at least two linear equations containing the same set of variables. (p.70)
A linear equation in two variables (ax + by = c) forms a straight line; in three variables it forms a plane. (p.71)
Three types of solutions: one solution (planes intersect at one point), infinite solutions (planes intersect along a straight line), no solution (planes do not intersect at any point). (p.73)
Solve by elimination or substitution, similar to simultaneous equations in two variables. (p.73)
Elimination: eliminate the same variable from two pairs of equations to get two equations in two variables, solve, then back-substitute. (p.74)
Result 0 = 0 means infinite solutions; result like 0 = 12 means no solution. (p.75)
Problem solving: define variables, form three equations, solve, then verify the answer. (p.77)
3.2 Simultaneous Equations Involving One Linear and One Non-Linear Equation
The solution is the point(s) of intersection of the graphs of the two equations. (p.80)
Methods: elimination, substitution (make one variable the subject of the linear equation), or graphical representation. (p.80)
After substitution, solve the resulting quadratic equation by factorisation, formula or completing the square. (p.80)
Graphical method: construct a table of values, plot both graphs; the intersection points are the solutions. (p.81)
Reject solutions that are not physically valid (e.g. negative lengths) in word problems. (p.83)