2x+y+8=12 & (I) 2x+y+4=8 & (II)
Note that both the x- and the y-variables have the same coefficient in both equations. Thus, in order to use the Elimination Method we will subtract Equation (II) from Equation (I).
We arrived to a true statement. Its truthfulness does not depend on any variable. Thus, any value for x and y will satisfy the statement. Hence, there are infinitely many solutions. This means the two lines are coincidental. Note that whenever we arrive to a true statement, the system has infinitely many solutions.
b
To answer this question, we will consider another example.
2x+y=8 & (I) - 2x+2y=- 8 & (II)
This time the x-variable has opposite coefficients. Thus, we will add the equations to eliminate it.
W found that the solution of the system, which is the point of intersection of the lines, is (4,0). Therefore, there is only one solution. Note that whenever we arrive to a concrete solution for each of the variables, the system has one solution.
c
Finally, to answer this question, we will use a third example.
x-y=4 & (I) x-y=0 & (II)
Note that the x- and the y-variables have the same coefficient in both equations. Therefore, to solve the system, we will subtract Equation (II) from Equation (I).
Since we know that 0≠4, we have arrived to a false statement. There are no values for x or y that satisfy it. Thus, the system has no solution. This means the lines do not intersect. Every time we arrive to a false statement, the system has no solution.
