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 Description  
 System of Equations:  Two or more equations that together define a relationship between variables usually in a problem situation. It can have no solution, one solution, or many solutions.

The solutions to a system of equations are called simultaneous because they satisfy all the equations in the system.

Consider this system of equations:

y = 3x + 4

y = -5x + 8

We can find the solutions to both equations by subtracting one equation from the other:

y = 3x + 4

-y = 5x - 8

0 = 8x - 4

=> 4 = 8x

=> x = 4/8 or 1/2 or 0.5

Substituting this value for x into one of the equations tells us that

y = 3(0.5) + 4 = 5.5.

Just to check, substitute the value for x into the other equation too:

y = -5(0.5) + 8 = 5.5

The fact that we got the same y-value using each equation is a good sign. It means that when x = 0.5, the y-values of both of these equations are the same, 5.5. Therefore, the simultaneous solution for this system of equations is (0.5, 5.5).

As another example, let's look at the system of equations in which one equation is linear and the other one is quadratic.

Again, in this situation we can find the solutions to both equations by subtracting one equation from the other:

 

Substituting these values in for x into one of the equations (and into the other just to check) tells us that when x = 1.22, y = 1.22, and when x = -1.22, y = -1.22. So, the simultaneous solutions for this system of equations are (1.22,1.22) and (-1.22, -1.22).