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MTH 119 — Spring — 2005 Essex County College — Division of Mathematics Test # 1^1 — Created February 22, 2005
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Show all work clearly and in order, and box your final answers. Justify your answers alge- braically whenever possible. You have at most 80 minutes to take this 100 point exam, each question is worth 10 points. No cellular phones allowed.
- Given the following linear system
2 x − y + 4 z = − 3 x − 2 y − 10 z = − 6 3 x + 4 z = 7 (a) Write the augmented matrix form of this linear system. 7.5 points
Solution: (^)
(b) Write the matrix equation form of this linear system. 7.5 points
Solution: (^)
x y z
- Solve for x: log 3 (2x − 1) − log 3 (x − 4) = 2 15 points
Solution: log 3 (2x − 1) − log 3 (x − 4) = 2
log 3
2 x − 1 x − 4
2 x − 1 x − 4
2 x − 1 = 9 (x − 4) 2 x − 1 = 9 x − 36 35 = 7 x 5 = x (^1) This document was prepared by Ron Bannon using LATEX 2ε.
Checking, to make sure that x = 5 is a solution is an important step.
log 3 (2 · 5 − 1) − log 3 (5 − 4) = 2 log 3 9 − log 3 1 = 2 log 3 9 = 2
So, x = 5 is a solution.
- Solve for x and y, using the inverse of the coefficient matrix. 15 points { 2 x + y = 1 3 x − y = 4
Solution: First write the matrix equation. [ 2 1 3 − 1
]
[
x y
]
[
]
Now find the inverse^2 of the coefficient matrix. [ 2 1 1 0 3 − 1 0 1
]
[
]
[
]
[
]
Finally, just interchange the rows to get the proper form. [ 1 0 1 / 5 1 / 5 0 1 3 / 5 − 2 / 5
]
Now use this inverse to solve the matrix equation for x and y. [ 2 1 3 − 1
]
[
x y
]
[
]
[
]
[
]
[
x y
]
[
]
[
]
[
x y
]
[
]
So the solution is x = 1 and y = −1. You should of course check this. { 2 − 1 = 1 3 + 1 = 4
There’s two values to place on the number line, 0 and 2. The analysis gives this solution: (−∞, 0) ∪ [2].
(b) x^3 > x 10 points
Solution:
x^3 > x x^3 − x > 0 x
x^2 − 1
x (x − 1) (x + 1) > 0
There’s three values to place on the number line, 0 and ±1. The analysis gives this solution: (− 1 , 0) ∪ (1, ∞).
(c) | 3 − 2 x| ≤ 5 10 points
Solution:
| 3 − 2 x| ≤ 5 − 5 ≤ 3 − 2 x ≤ 5 − 8 ≤ − 2 x ≤ 2 4 ≥ x ≥ − 1
The final inequality is equivalent to − 1 ≤ x ≤ 4: [− 1 , 4].
