Math 2270
Quiz 1
1. Use elimination to find all solutions of the system
2x + 4y = 3
3x + 6y = 2
.
If we multiply the first equation by 1/2, the system becomes
x + 2y = 3
2
3x + 6y = 2
.
Next, we multiply the first equation by −3 and add it to the second to get
x + 2y = 3 2 .
0 = −5
2
In this form, the second equation tells us that there is no solution to the system.
2. Find all the polynomials f (t) of degree ≤ 2 whose graph runs through the points (1, 3) and (2, 6) such that
f ′ (1) = 1.
Any polynomial of degree less than or eqqual to 2 can be written as f (t) = at2 + bt + c for appropriate choices
of a, b, and c. The information in the problem tells us that f (1) = 3, f (2) = 6 and f ′ (1) = 1. Of course,
f ′ (t) = 2at + b. Applying these conditions gives rise to the system
a=2 a − 1c = 0 a+b+c=3 a+b+c=3 2
4a + 2b + c = 6 → −2b − 3c = −6 → b + 3 c = 3 → b = −3 .
1 2
c=4 − c = −2 −b − 2c = −5 2a + b = 1
2
Therefore, f (t) = 2t2 − 3b + 4.
1