Consider the two lines in R 2 defined by the equations 2x y = 2 and 1 3 x 2 3 y
ID: 2253036 • Letter: C
Question
Consider the two lines in R 2 defined by the equations 2x y = 2 and 1 3 x 2 3 y = 0. (a) Find the point of intersection between these two lines. (b) Consider a generic third line ax + by = c, where a, b, c are arbitrary numbers. Find conditions on a, b, c so that all three lines intersect in the same point. (Hint: You are essentially trying to figure out for which numbers a, b, c the system
has a solution. Modify the idea from example 5 in section 1.1 to find a condition on a, b, c for this system to be consistent.)
Consider the two lines in R2 defined by the equations 2z-,-2 and = 0. (a) Find the point of intersection between these two lines (b) Consider a generic third line az by = c, where a,b,c are arbitrary numbers. Find conditions on a,b, c so that all three lines intersect in the same point. (Hint: You are essentially trying to figure out for which numbers a, b, c the system 2x-y=2 az +by- c has a solution. Modify the idea from example 5 in section 1.1 to find a condition on a, b,c for this system to be consistent.)Explanation / Answer
(a) 2x-y = 2 -------(1)
x/3 - 2y/3 = 0 => x = 2y -----(2)
Put x = 2y in (1) we get , 2*(2y)-y = 2=> 4y-y=2 => 3y=2=> y = 2/3
x = 2y = 2*2/3 => x = 4/3
Point of intersection P is(4/3,2/3)
(b) If all the three lines intersect at the same point then ax+by=c must also pass through P
=> a*4/3 + b*2/3 = c => 4a+2b = 3c