Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 01 (c)
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Question B 01 (c) 

Consider the lines
(c) Do the lines and intersect? If so, find the intersection point. If not, explain. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

A point lies on if and only if for some . Similarly, iff for some . 
Hint 2 

Translate this problem into a system of linear equations problem. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. If a point is a intersection point of the lines and , then and . In other words, for some and . By expanding the second equality, we have The first component implies that and we plug this into the equation from the second component; So, we have and . However, these values of and doesn't satisfy the equation from the third component; This means there's no and to make hold. In other words, there's no intersection point. 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Translate this problem into a problem involving a system of linear equations. To do this, note that and intersect exactly when there are numbers and that satisfy
rearranging this gives
and
Writing as a system of linear equations gives,
In matrix form,
In other words, and intersect exactly when the above system of linear equations has at least one solution. Using Gaussian Elimination, we obtain
The second row of the last matrix reads
which is impossible. Hence, the system of linear equations has no solution and the lines and do not intersect. 