Question

Betty and Andrea have been solving systems of equations with one polynomial function of degree two and one linear function. Betty says there must always be two solutions, and Andrea says there will no solution. Using complete sentences, explain how Betty can be correct, how Andrea can be correct, and how they both can be wrong.

Answer 1

Since Quadratic equations have two solutions, in the system of equations polynomial function of degree two will have two numbers satisfying the condition. Also possible that the two solutions of quadratic equation are also the solution of the linear function. Thus, thinking so Betty is correct.
It is also possible that the values satisfying the quadratic equation may not satisfy the Linear equations, Hence the system may have no solutions satisfying both equations making Andrea's answer correct
The remaining one possibility is that only one solution of the quadratic equation satisfy the linear function thus the systems having only one common solution. In that case,both are wrong.
In short if the graph of two functions cut each other at two points then Betty is correct
If both the graphs don't intersect then Andrea is correct and if both the graph intersects at a single point or touch each other then both are wrong.