Question

In attempting to solve the system of equations and , John graphed the two equations on his graphing calculator. Because he saw only one line, John wrote that the answer to the system is the empty set. Is he correct? Explain your answer.

Answer 1

The correct answer is NO.

Explanation:

John is graphing two lines with equations say ax + by + c = 0 and dx + ey +f =0 on his graphing calculator.

For John to get a unique solution the coefficients should follow the following condition
`(a)/(d) \\eq (b)/(e)`. So when he plots the lines he get an intersecting pair of lines.

For John to get no solution the coefficients should follow the following condition
`(a)/(d) = (b)/(e) \\eq (c)/(f)`. When we plot in this case we get two parallel lines.

For John to get an infinite many solution the coefficients should follow the following condition
`(a)/(d) = (b)/(e) = (c)/(f)`. And when we plot the lines in this case we get one line superimposed on the other.

Since after graphing, John sees only one line, the lines must have superimposed on one another giving John an infinite number of solutions.