Simultaneous Equations

Fundamental Understanding

A system of simultaneous equations includes two or more equations with the same set of variables.
Solutions to these equations are the values which satisfy all the equations in the system at the same time.

Solution Methods

To solve simultaneous equations, one can use substitution, elimination, or graphical methods.
Substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination method involves adding or subtracting the equations in order to eliminate one of the variables.
Graphical method involves drawing the graphs of the equations on the same set of axes. The points where the graphs intersect represent the solutions.

Quadratic Simultaneous Equations

A set of simultaneous equations is quadratic if at least one of the equations is quadratic.
When solving quadratic simultaneous equations, either substitution or elimination can be used, but often substitution method is easier.
After replacing the variable from one equation into another, a quadratic equation should result, which can be solved using quadratic formula, factorisation or completing the square method.

Real-World Applications

Simultaneous equations are frequently used in real-life problems and situations.
They are found in physics, business and economics, engineering, statistics and various other scientific and non-scientific areas.

Remember, practice is key when it comes to solving simultaneous equations. Not only should you understand the concept theoretically, but you should also solve problems involving various types of simultaneous equations.