1. PROGRAMME F5
LINEAR EQUATIONS and
SIMULTANEOUS LINEAR EQUATIONS

2. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Simultaneous linear equations with three unknowns

3. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Simultaneous linear equations with three unknowns

4. Programme F5: Linear equations and simultaneous linear equations
Solution of simple equations
A linear equation in a single variable (unknown) involves powers of the variable no higher than the first. A linear equation is also referred to as a simple equation.
The solution of simple equations consists essentially of simplifying the expressions on each side of the equation to obtain an equation of the form:

5. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Simultaneous linear equations with three unknowns

6. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Solution by substitution
Solution by equating coefficients

7. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Solution by substitution
A linear equation in two variables has an infinite number of solutions. For two such equations there may be just one pair of x- and y-values that satisfy both simultaneously. For example:

8. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Solution by equating coefficients
Example:
Multiply (a) by 3 (the coefficient of y in (b)) and multiply (b) by 2 (the coefficient of y in (a))

9. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with two unknowns
Simultaneous linear equations with three unknowns

10. Programme F5: Linear equations and simultaneous linear equations
Simultaneous linear equations with three unknowns
With three unknowns and three equations the method of solution is just an extension of the work with two unknowns.
By equating the coefficients of one of the variables it can be eliminated to give two equations in two unknowns. These can be solved in the usual manner and the value of the third variable evaluated by substitution.

11. Programme F5: Linear equations and simultaneous linear equations
Pre-simplification
Sometimes, the given equations need to be simplified before the method of solution can be carried out. For example, to solve:
Simplification yields:

12. Programme F5: Linear equations and simultaneous linear equations
Learning outcomes
Solve any linear equation
Solve simultaneous linear equations in two unknowns
Solve simultaneous linear equations in three unknowns