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