# How to Solve Special Systems

A special system of $$2$$ equations is a system of equations that can be solved using a specific method, such as substitution or elimination. Two common types of special systems are Consistent, dependent systems and Consistent, independent systems.

## Step-by-step to solve special systems

To solve special systems, follow the step-by-step guide below:

1. Consistent, dependent systems: A consistent, dependent system of $$2$$ equations is a system of equations that has an infinite number of solutions. This occurs when the two equations are identical or one equation is a multiple of the other. For example, the system of equations $$2x + 3y = 6$$ and $$4x + 6y = 12$$ has an infinite number of solutions because the second equation is simply twice the first equation. In this case, we can solve the system by solving either equation for one variable and then substituting it into the other equation.
2. Consistent, independent systems: A consistent, independent system of $$2$$ equations is a system of equations that has one unique solution. This occurs when the two equations are not identical and the system of equations can be solved by elimination or substitution.

### Solving Special Systems– Example:

Solve the system of equations

$$\begin{cases}2x+3y=6\\ 4x+6y=12\end{cases}$$

Solution:

Using elimination method: Add the two equations $$\begin{array}{r} &2x + 3y = 6\\ +\!\!\!\!\!\!&4x + 6y = 12 \\ \hline \end{array} \\ \ \ \ \ \ \ 6x + 9y = 18$$

So, we can see that $$x = -3$$ then we can substitute this value in one of the equations

$$2(-3) + 3y = 6$$→$$y = 3$$

So, the solution of this system is $$x = -3$$ and $$y = 3$$

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