Question

What are the solutions to the following system of equations? y=x^2+12x+30 and 8x-y=10

Answer 1

Note that the 2nd equation can be re-written as y=8x-10.
According to the second equation, y=x^2+12x+30.
Equate these two equations to eliminate y:

8x-10 = x^2+12x+30

Group all terms together on the right side.  To do this, add -8x+10 to both sides.  Then 0 = x^2 +4x +40.  You must now solve this quadratic equation for x, if possible.  I found that this equation has NO REAL SOLUTIONS, so we must conclude that the given system of equations has NO REAL SOLUTIONS.

If you have a graphing calculator, please graph 8x-10  and  x^2+12x+30 on the same screen.  You will see two separate graphs that do NOT intersect.  This is another way in which to see / conclude that there is NO REAL SOLUTION to this system of equations.