Question

danny and gina both solved the following problem. is either of them correct? explain any mistakes they made and what the correct answer is. the diagram shows two parallel lines that are cut by a transversal.
danny’s work:
$6x - 30 = 60$
$6x = 90$
$x = 15$
gina’s work:
$6x - 30 + 60 = 180$
$6x - 30 = 180$
$6x = 210$
$x = 35$
gina is completely correct.
neither of them are correct. their work is wrong and they set up the problem incorrectly.
danny is completely correct.
neither of them are completely correct. danny set up the problem wrong, but his work is correct. gina set up the problem correctly, but her work is wrong.

Explanation:

1. Analyze Angle Relationship: The two angles \(6x - 30\) and \(60^\circ\) are same - side interior angles (since the lines are parallel and cut by a transversal), so they should be supplementary, meaning \(6x-30 + 60=180\). So Gina's setup \(6x - 30+60 = 180\) is correct, but when she simplified \(6x-30 + 60=180\) to \(6x - 30=180\), she made a mistake (she should have \(6x+30 = 180\) after combining like terms).
2. Analyze Danny's Work: Danny set up the equation \(6x - 30=60\), which is incorrect because the angles are not equal (they are supplementary). However, if we consider his equation \(6x - 30 = 60\), his solving steps are correct: adding 30 to both sides gives \(6x=90\), and then dividing by 6 gives \(x = 15\). But the setup of the equation is wrong because of the incorrect angle relationship assumption.
3. Conclusion: Since Danny's setup is wrong (even though his solving is correct) and Gina's solving is wrong (even though her setup is correct), neither of them is completely correct.

Answer:

Neither of them are completely correct. Danny set up the problem wrong, but his work is correct. Gina set up the problem correctly, but her work is wrong.