(KS4, Year 10)
The LessonSimultaneous equations are a set of several equations with several unknowns. There are a set of values of these unknowns which solve all the equations at the same time.
Dictionary DefinitionThe Merriam-Webster dictionary defines simultaneous equation as "satisfied by the same values of the variables."
A Real Example of Simultaneous EquationsIt is easier to understand simultaneous equations with an example. The most common type of simultaneous equations are a set of 2 linear equations, where the unknowns are the variables x and y:
- Both of these equations are linear. They show a straight line when plotted on a graph.
- Both of these equations have two unknowns (or variables) : x and y. They can take different values.
- Each equation has different coefficients and constants. In the top equation, the coefficient of x is 2. The coefficient of y is 1. The constant is 4. In the bottom equation, the coefficient of x is 1. The coefficient of y is 3. The constant is 7.
How to Solve Simultaneous EquationsThe solution to simultaneous equations are the values of x and y which solve the equations simultaneously. The solution to the example above is x = 1 and x = 2:
Read more about solving simultaneous equations
More Real Examples of Simulaneous EquationsSimultaneous equations do not have to be 2 linear equations.
3 (or More) Simultaneous EquationsIt is possible for 3 or more linear equations to be solved simultaneously. 3 equations with 3 unknowns (x, y and z) are shown below. There are as many equations as there are unknowns (see Note.)
These simultaneous equations are solved by x = 1, y = 2 and z = 3.
Linear and Quadratic EquationsIt is possible for one equation to be linear and another to be a quadratic equation.
Because of the quadratic equation, there will be 2 pairs of values of x and y that solve the equations simultaneously. These equations are solved by x = 0, y = 0 and x = 1, y = 1.
What's in a Name?Simultaneous equations are so called because the solution solves all the equations at the same time, or simultaneously.
Number of Unknowns = Number of EquationsWhen there are 2 sets of linear equations:
- there are 2 equations.
- there are also 2 unknowns (x and y).
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