Expert: Chanda Walker - 3/16/2009

Question
QUESTION: Solve the following pairs of simultaneous equations. If there is not a single solution state how many solutions there are?

(i)      2x + y = 3
     x – 2y = -1

(ii)      -x + 2y = 8
     5x – y = -15
  
(iii)   2x + 6y = 8
     4x + 12y = -5

(iv)      2x – 2y = -4
     -3x + 3y = 9

(v)      - ½x + ½y = 1
     5x – 5y = -10


ANSWER: 2x + y = 3
x – 2y = -1

**************

x = 2y - 1
2(2y -1) + y = 3
4y -2 + y = 3
5y = 5
y = 1

x = 1

****************************************
-x + 2y = 8
5x – y = -15
*************
x = 2y - 8
5(2y - 8) - y = -15

10y -40 -y = -15
9y = 25
y = 25/9
y = 2 7/9

x = -22/9
x = -2 3/9

******************************************
2x + 6y = 8
4x + 12y = -5
*************
x =4 - 3y
4(4-3y) + 12 y = -5
16 - 12y + 12y = -5
16 does not equal -5

There is no solution to this.  These equations are inconsistent.
******************************************
2x – 2y = -4
-3x + 3y = 9
************
Again, these solutions are inconsistent.  There are no solutions.

******************************************
- ½x + ½y = 1
5x – 5y = -10

There are an infinite number of solutions as long as x = y -2.


---------- FOLLOW-UP ----------

QUESTION: Hi, can you please help me with these questions please, I've been trying to solve them for 4 months now. ( just a hobby)

1. Using the following parametric representation draw the following curve:
     
What closed curve do you obtain?      

2. Calculate the dot product of each of the following pairs of vectors, and also state what the result means about the angle between the vectors:

(i)      (i + j ). (2i +3j)

(ii)      (2i + 6j ). (4i + 12j)

(iii)   (3i – j + 2k ). (3i + 3j - 3k)

(iv)      (i - 4j - k ). (4i + j + k)

(v)      (3.6i + 8.2j – 6.5k ).( 2.2i + 7j + 2k)

3. If the displacement of a particle P at time t is given by  , find it’s (a) velocity when t=2s, t=5s and t=7s  (b) its acceleration at t=4s and t = 5s
         
4. Write a C++ program that determines   using the mid-ordinate rule and the trapezoidal rule. The number of ordinates (or rectangles or trapezia) should be entered at runtime by the user. Decide before you run the program what f(t) is to be and use this in a function call, do not worry about converting a string in your function.

Compare your results using these programs with the following exact integral:  

,

use 10 trapezia and 10 ordinates (in the mid-ordinate rule). Increase the number of trapezia to:
i) 100
ii)1000.


ANSWER: There is no question in the first one.  I'm just starting the others now.  I will edit this answer but I wanted to catch you while you were still on line, hopefully.

(i + j).(2i + 3j)

{1,1}.{2,3} = 1*2 + 1*3 = 5

{2, 6}.{4, 12} = 4*2 + 6*12 = 8 + 72 = 80

That is how these sorts of problems are worked.

3.  This has no equation.

4.  I don't use C++.



---------- FOLLOW-UP ----------

QUESTION: If the displacement of a particle P at time t is given by (LOOK AT IMAGE FOR THIS PART OF SENTENCE IN ATTACHMENT), find it’s

(a) velocity when t=2s, t=5s and t=7s  

(b) its acceleration at t=4s and t = 5s

Answer

s'(t) = 3Sqrt(t)  - 1/(2Sqrt(t)) + 3  this is the velocity

s''(t) =  (3/(2 Sqrt(t)) + 1/(4 t^(3/2))  this is the acceleration

Just plug the values in for t and you'll have your answers.

After being asked so many questions, I feel it is only fair to just get you started on these problems instead of solving them all for you.

If you want further help, please post the part of the solution you have achieved so far and I'll pick up at your mistake--or I'll tell you that you have it all correct.

I'm sorry for any inconvenience.  I love TEACHING people math and helping them. Please understand.