  Assignment 21

 "The Great Architect of the Universe now begins to appear as a pure mathematician." -- (James Jeans) Check out these passages from the Bible: I Kings 7:23 and II Chronicles 4:2. You will find indirect references to indicate that pi is equal to 3 at each location if you know that the circumference of a circle is given by the formula C = 2(pi)R. The passages contain these words and phrases: cubit : Originally, this was roughly the distance from one's elbow to the end of middle finger. In English measure, it is about 18 inches. molten sea : A high bowl or tank supported on twelve oxen. (Biblical description appears in I Kings 7:24-26.) bath : A liquid measure, approximately 6 gallons. What does Herkimer's call the underwear worn by a mermaid who likes mathematics? Answer: Algebra. Herky 's friends: DON KEY...this guy raised pack animals. LOTTA RAIN ...this lady loves Seattle. ASSIGNMENT #21 Reading: Section 3.6, pages 177-180. Written: Page 181/12,13,15,18,22 Practice communication. Be neat and organized in your presentations. Mathematical word analysis:FUNCTION: From the Latin word functio (to perform). If f(x) = 2x + 7, then f is a rule that has you "perform" the operations multiplication and addition.

This is where neatness and organization really pay off. We are solving systems of 3 equations with 3 unknowns. An equation like 3x - 2y + 5z = 7 represents a plane in 3-D space. When you solve a 3-by-3 system such as the ones in your homework, you are finding the points of intersection of three planes. Your intuition should tell you that three distinct planes can do the following:

• Intersect in a single point. (Think of the upper right corner of a room, for instance.)
• Intersect in a line. (Think of two walls intersecting in a line, and a third plane intersecting that line.)
• Not intersect at all. (Think of the planes of the ceilings in multi-level buildings.)

Example of solving a 3-by-3 system (Note the communication):

(a): -2x + y + 3z = -8
(b): 3x + 4y - 2z = 9
(c): x + 2y + z = 4

Strategy: First, eliminate the variable y.

(d) = 4(a): -8x + 4y + 12z = -32.

(e) = (b) - (d): 11x - 14z = 41.

(f) = 2(c): 2x + 4y + 2z = 8.

(g) = (b) - (f): x - 4z = 1.

(g) ==> x = 4z + 1.

Substituting into (e) yields 11(4z+1) -14z = 41 ==> 44z+11-14z = 41 ==> 30z = 30 ==> z = 1.

(g) ==> x = 4(1) + 1 = 5.

(c) ==> 5 + 2y + 1 = 4 ==> 2y = -2 ==> y = -1.

Conclusion: The three planes (a), (b), and (c) intersect at (5,-1,1).

PATIENCE IS A VIRTUE.
MATH IS POWER.

 Problem: Solve the system (a): x + y = 22(b): x + y + z = 18(c): x + z = 16 Solution (with communication): Substituting (a) into (b) yields 22 + z = 18 ==> z = -4. (c) ==> x + (-4) = 16 ==> x = 20. (a) ==> 20 + y = 22 ==> y = 2. The 3 planes intersect at the point (20,2,-4). Problem: Solve the system (a): 2x + 3y + z = 11(b): x + 4y - z = 9(c): 5x + y + z = 13 Solution (with communication): (d) = (a)+(b): 3x + 7y = 20 (e) = (a) - (c): -3x + 2y = -2 (f) = (d)+(e): 9x = 18 ==> y = 2. (d) ==> 3x + 14 = 20 ==> 3x = 6 ==> x = 2. (a) ==> 4 + 6 + z = 11 ==> z = 1. The 3 planes intersect at the point (2,2,1).