There are still the special cases to consider.
There are still the special cases to consider.
Source: Systems of Equations, wallacemath, You Tube
Again, this condition is also true for substitution.
Source:Special Cases with Substitution, wallacemath, You Tube
As you saw in the video clips, if both variables are eliminated and you get a false statement, such as 0 = any number other than 0, the equations in the system represent parallel lines. This condition is true for substitution as well as elimination. When you get a true statement when both variables are eliminated, where a number is equal to itself, such as 0 = 0, then the equations in the system represent the same line. The entire line would be the intersection and since it is made up of infinitely many points, the solution to the system has infinitely many points. Again, this condition is also true for substitution.
Let’s now look at systems with a non-linear equation.
Solve using elimination: 
Solution:
The only variable that we have in both equations that could possibly be eliminated is y. Because we have both a quadratic and a linear term of x in one equation but not the other, it would be impossible to eliminate x.
Notice that both equations have a subtraction sign in front of y, so we will need to multiply one of the equations by -1 so that each y-term will have opposite signs. We want a positive x2, so you’ll multiply the first equation by the -1.
| x2 − 6x = –9 | Add the equations |
| x2 − 6x + 9 = 0 | Add 9 to both sides of the equation so the quadratic equation equals 0 |
| (x − 3)2 = 0 | Factor |
| x = 3 | Solve the factor for x |
| 2(3) − y = 6 | Substitute 3 in for x in the linear equation |
| 6 − y = 6 | Simplify |
| –y = 0 | Subtract 6 from both sides of the equation |
| y = 0 | Divide both sides of the equation by -1 |
The solution is (3,0).