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H. Exercises to Try
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Solving Linear
Equations
(Feb 14, 2005)
with & then without stick diagrams plus   solving word problems; and solving systems: -  essentially one unknown, essentially triangular & general

 

A. Letters and Lengths
B. Solving Linear Eq'ns - WS
C. Solving Linear Eq'ns - No Sticks
D. Almost One
E Substitution Methods
F. Two More Methods
G. Larger Systems
H.  Exercises to Try


Read them in order

(i) x + 20 = 29 WS
(ii) 2x + 6 = 24 WS
(iii) 3x + 10 = 32 WS
(iv) 5a + 16 = 3a+ 24 WS
(v)  (½)x + 8 = 24½ WS
(vi)  (¾)a + 16 = (¼)a+ 24 WS
(vii) (¾)q + 17 = 32 WS
(viii) 13 =[2/3]x +7 twice WS
(5/6)q + 8+(5/6) = 14 + (2/3) WS
Proper Use of Equal Sign


Up
A. Letters and Lengths
B. Solving Linear Eq'ns - WS
C. Solving Linear Eq'ns - No Sticks
D. Almost One
E Substitution Methods
F. Two More Methods
G. Larger Systems
H.  Exercises to Try
 

 

Exercises

Instruction:  If your method is correct, but your check says your answer is wrong, writes OOPs besides your check, and go on. Correct your work later. The error will be somewhere between the start of your solution and the end of the check.

1A. (4 points) Express A and B as simplified, improper fraction:  For the calculation of A, cancel common factors in numerators and denominators before multiplication. For the calculation of B, use the least common denominator.

A =  25
27
. 81
100
 B =   3
26
 + 7
10

For A and B, there is 1 point for right answer and 4 points for following instruction on how to arrive at the answer. Show work!  If you use a calculator (not recommended), show the reasoning needed to arrive at your solution without its use.

2 . (6 points) Express as a single, simplified, improper fraction:

A= [ 5
4
 ¸ 35
24
] ¸ 2
7
         
B=  4
5
[ 5
4
 + 35
24
 + 2
7
]
C =  8
15
[ 25
4
 + 75
8
 + 15
2
]

3A.   (4 points) Solve the (scrambled) triangular system

3x +  2y +  z = 65
      3y      = 45
6x +   y      = 75

3B.  (4 points) Solve the (scrambled) triangular system

2x +  3y + 4z = 21
      8y + 2z = 26
      5y      = 15

 4. (Bonus 2 points) John has twice as many apples as Harry.  Harry and John together have 18 apples.  Introduce notation for the number of apples each has. Write down a system of two equations in two unknowns that would allow you to find the number of apples each has. Do not solve the system.

5.  (a)   (8 points)  Solve for x and y  - obtain a literal solution.

 ax      = p
 bx + cy = q

     (b) (2 points) Verify (check) your solution.

(c)(2 points)  Use your formulas for the solution in (a) to solve

 4x      = 10
 2x + 2y = 8

6. (12 points)  Solve 

3x + 2y = 17
2x -  y =  2

7. (10 pointsSolve  

  x - 2y = 4
-3x +  y = 3

8.  (12 points)  Solve  

 x +  y +  z= 12
2x + 3y + 5z = 32
6x + 7y +  z =64

9.  (10 pointsSolve  - note minus sign in each equation.

 -x +  y +  z = 10 
   x  - y +  z = 8
   x +  y -  z = 6
 x +  y +  z - w =14

10.  (4 points)  (i)   Show by substitution that the formulas

x =   dg
ad-bc
 and  y =  -cg
ad-bc

          satisfy (solve) the system

 ax + by = g
 cx + dy = 0

     (2 points) (ii)  Give values  for coefficients a, b, c, d, and g  to make the system in (i) the same as 

 4x + y  = 10
  x + 4y = 0

  (4 points) (iii) Use the formulas in (i) to solve the system in (ii)

 

 

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3 .Why.Slopes.&
.More.Math.1995

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5.  Proportionality, Back- & For-wards - theme at work.
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16   Calculus - Examples & Advice 
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19 Maps, Similarity & Trig, (alt view)
20 Complex numbers  

21 Logic with Symbols+truth tables

22  Consistent Story Telling
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