If you were solving a system of equations and you came to a statement like 4 = 4, what do you know about the system?

Question

whpalmer4 · Answer

Let's look at an example:

$$x+y = 1$$$$-x-y=-1$$

If we add the two equations together, or use substitution, after the dust settles, we're left with $$0=0$$

Now, observe that if we take the first equation and multiply it by -1, we get the second equation.    What does that mean?  Well, it means that the two lines are identical.  A solution is found wherever the same point satisfies all the equations in a system, so any point on the line is a solution. There are infinitely many points on the line, so we have infinitely many solutions.  If you solve both equations for $y$, you'll see that you do have the same formula each time.

Let's look at another very closely related example:
$$x+y =1$$$$x+y=0$$Here if we solve each equation for $y$, we get respectively$$y= -x+1$$$$y=-x$$

If we think about slope-intercept form $(y=mx+b)$, both lines have the same slope, but a different y-intercept.  That means they are parallel, and parallel lines do not intersect.  What happens if we solve the system?

I'll solve the system by plugging in the expression for $y$ from the second solution in the first equation:
$$x + y = 1$$$$x + (-x) = 1$$$$0 = 1$$
When we get a result like this, it means the lines have the identical slope but have different y-intercepts (parallel), distinguishing from the case of lines with the identical slope and identical y-intercepts (coincident).

To sum up, there are three cases:

1) 0 = 0 
Lines are coincident, infinitely many solutions
2) 0 = 1 (or any number other than 0)
Lines are parallel, no solutions
3) (x,y) (for some value of x and y)
Lines intersect at a point; that point is a solution to the system