Initialise
Clear the workspace and load Linear Algebra package
> restart;
> with(LinearAlgebra):
If you want practice at hand calculation you should use the worksheet "Interactive Gaussian Elimination" (see Menu) Enter the matrix of coefficients and right-hand side vector You may edit the following statements or use the matrix and vector pallettes to enter new data ( see View, Palettes) > A:=; > b:=;

Form the augmented matrix and solve
The Maple routine GaussianElimination requires the augmented matrix A|b as input. In this worksheet this matrix is called Ab and is formed using > Ab:=;
The row echelon form H|c (here called Hc) of Ab is computed by GaussianElimination. (For a square system of equations the row echelon form of A is upper triangular ) Note that the Maple function GaussianElimination performs a systematic version of the idea of elementary row operations: (i) Multiples of R1 are subtracted from R2, R3 . . Rn to reduce the elements below the leading diagonal in the first column to zero. (ii) In the resulting system multiples of R2 are subtracted from R3, R4 . . Rn to reduce the elements below the leading diagonal in the second column to zero. (iii) This process is continued to produce a system in row echelon form, using essential row interchanges where necessary. > Hc:=GaussianElimination(Ab);

Look at the form of H|c before executing the next statement. Is there a unique solution, infinitely many solutions or no solution ? The solution vector, x, is found from H|c using BackwardSubstitute x:=BackwardSubstitute(Hc);

Maple techniques: Accessing vector components and matrix elements and checking that Ax = b The components of x are x[1], x[2] ....:
> x[1];
> x[2];

The elements of A are A[i,j], for example:
A[2,1];

...Patterns within systems of LinearEquations
HL Type 1 Maths Coursework
Maryam Allana
12 Brook
The aim of my report is to discover and examine the patterns found within the constants of the linearequations supplied. After acquiring the patterns I will solve the equations and graph the solutions to establish my analysis. Said analysis will further be reiterated through the creation of numerous similar systems, with certain patterns, which will aid in finding a conjecture. The hypothesis will be proven through the use of a common formula. (This outline will be used to solve both, Part A and B of the coursework)
Part A:
Equation 1: x+2y= 3
Equation 2: 2x-y=4
Equation 1 consists of three constants; 1, 2 and 3. These constants follow an arithmetic progression with the first term as well as the common difference both equaling to one. Another pattern present within Equation 1 is the linear formation. This can be seen as the equation is able to transformed into the formula ‘y = mx+c’ as it is able to form a straight line equation (shown below). Similar to Equation 1, Equation 2 also follows an arithmetic progression with constants of; 2, -1 and 4. It consists of a starting term of 2 and common difference of -3. As with Equation 1,...

...2014/9/16
LinearEquations
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LinearEquations
A linearequation is an equation for a straight line
These are all linearequations:
y = 2x+1
5x = 6+3y
y/2 = 3 x
Let us look more closely at one example:
Example: y = 2x+1 is a linearequation:
The graph of y = 2x+1 is a straight line
When x increases, y increases twice as fast, hence 2x
When x is 0, y is already 1. Hence +1 is also needed
So: y = 2x + 1
Here are some example values:
http://www.mathsisfun.com/algebra/linear-equations.html
x
y = 2x + 1
1
y = 2 × (1) + 1 = 1
0
y = 2 × 0 + 1 = 1
1
y = 2 × 1 + 1 = 3
1/6
2014/9/16
LinearEquations
2
y = 2 × 2 + 1 = 5
Check for yourself that those points are part of the line above!
Different Forms
There are many ways of writing linearequations, but they usually have constants (like "2" or
"c") and must have simple variables (like "x" or "y").
Examples: These are linearequations:
y = 3x 6
y 2 = 3(x + 1)
y + 2x 2 = 0
5x = 6
y/2 = 3
But the variables (like "x" or "y") in LinearEquations do NOT have:
Exponents (like the 2 in x2)
Square roots, cube roots, etc
Examples: These are NOT linearequations:
...

...Summer 2010-3 CLASS NOTES CHAPTER 1
Section 1.1: LinearEquations
Learning Objectives:
1. Solve a linearequation
2. Solve equations that lead to linearequations
3. Solve applied problems involving linearequations
Examples:
1. [pic]
[pic]
3. A total of $51,000 is to be invested, some in bonds and some in certificates of deposit (CDs). If the amount invested in bonds is to exceed that in CDs by $3,000, how much will be invested in each type of investment?
4. Shannon, who is paid time-and-a-half for hours worked in excess of 40 hours, had gross weekly wages of $608 for 56 hours worked. What is her regular hourly wage?
Answers: 1. [pic]
2. [pic]
3. $24,000 in CDs, $27,000 in bonds 4. $9.50/hour
Section 1.2: Quadratic Equations
Learning Objectives:
1. Solve a quadratic equation by (a) factoring, (b) completing the square, (c) the
quadratic formula
2. Solve applied problems involving quadratic equations
Examples:
1. Find the real solutions by factoring: [pic]
2. Find the real solutions by using the square root method: [pic]
3. Find the real solutions by completing the square: [pic]
4. Find the real solutions...

...Algebra I Chapter 5 Study Guide Writing LinearEquations
Name ________________
Due: Tuesday, January 17 (Exam week)
100 points
Writing LinearEquations in a Variety of Forms
Using given information about a __________, you can write an ________________of the line in _____________ different forms. Complete the chart:
Form (Name)
Equation
• •
Important information
The slope of the line is ____. The __ - ___________ of the line is _____. The slope of the line is _____. The line passes through ( ______, ______ ) A, B, and C are __________ numbers. A and B are not both ___________.
Slope – Intercept
Point – Slope
• •
Standard
• •
Try a few (Page 345 – 347) Write an equation in Slope – Intercept Form:
Algebra I
Study Guide Chapter 5
lopeWrite an equation in Slope-Intercept Form that passes through the given point and has the given slope m.
8. (23, 21); m = 4 y= m= x= b= 9) (– 2, 1), m = 1 y= m= x= b= 10) (8, –4) m = – 3 y= m= x= b=
Write an equation in Point-Slope Form that passes through the given points. oint11) (4, 7) (5, 1) 12) (9, 22) (23, 2) 13) (8, 28) (23, 22)
(Hint: you need slope)
(Hint: you need slope)
(Hint: you need slope)
-2-
Algebra I
Study Guide Chapter 5
Write an equation in Standard Form of the line that has the given characteristics.
Hint for #15 & #16 You’ll...

...SYSTEM OF LINEAREQUATIONS IN TWO VARIABLES Solve the following systems: 1.
x y 8 x y 2
by graphing
by substitution
by elimination
by Cramer’s rule
2.
2 x 5 y 9 0 x 3y 1 0
by graphing
by substitution
by elimination
by Cramer’s rule
3.
4 x 5 y 7 0 2 x 3 y 11 0
by graphing
by substitution
byelimination
by Cramer’s rule
CASE 1: intersecting lines independent & consistent m1m2
CASE 2: parallel lines inconsistent m1 = m2 ; b1 b2
CASE 3: coinciding lines consistent & dependent m1 = m2 ; b1 = b2
Classify the following system, whether (a) intersecting, (b) parallel, or (c) coinciding lines 1.
3 x 4 y 1 0 3 x 4 y 2 0 3 x 4 y 1 0 6 x 8 y 2 0
Solve the following systems in three variables: 1. 3 x 4 y z 1 2. x y 2
x 4 y 3z 3 3 x 2 y 2 z 0
________
3 y z 1 x 2 z 7
2.
________
3.
2 x 5 y 1 0 5 x 2 y 2 0
2 x y 1 4 x 2 y 3 x 2 y 1 0 2 x y 1
________
4.
________
5.
________
1 x Solve 1 x
2 3 y 3 2 y
Problem solving Form a system of equations from the problems given below. A) (MIXTURE PROBLEM 1) How many pounds of a 35% salt solution and a 14% salt...

...1 ) The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.
2 ) A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed?
3 ) These circles are identical. What is the value of x ?
4 ) Solve for x using these two equations: 2x + 6 = y; y - x = 2
5 ) The perimeter and the area of this shape are equal. What is the value of x?
6) Shobo’s mother’s present age is six times Shobo’s present age . Shobo’s age five years from now will be one third of his mother’s present age . What are their present ages ?
7)There is a narrow rectangular plot , reserved for a school in A Mahuli village . The length and breadth of the plot are in the ratio 11:4 . At the rate of Rs. 100 per meter it will cost the village panchayat Rs. 75000 to fence the plot . What rare the dimensions of the plot ?
8)Solve the following equations and check your results :
1 . 3x = 2x + 18
2 . 5t-3 = 3t-5
3 . 5x + 9 = 5 = 3x
4 . 4z = 3 = 6 = 2x
5 . 2x- 1 = 14 – x
9 )The organizers of an essay competition decide that a winner in the competition gets a prize of Rs. 100 and a participant who does not win get a prize of Rs. 25 . The total prize money distributed is Rs . 3000 . Find the number of winners , if the total number of participants...

...CHAPTER 8
Linear Programming Applications
Teaching Suggestions
Teaching Suggestion 8.1: Importance of Formulating Large LP Problems.
Since computers are used to solve virtually all business LP problems, the most important thing a student can do is to get experience in formulating a wide variety of problems. This chapter provides such a variety.
Teaching Suggestion 8.2: Note on Production Scheduling Problems.
The Greenberg Motor example in this chapter is largest large problem in terms of the number of constraints, so it provides a good practice environment. An interesting feature to point out is that LP constraints are capable of tying one production period to the next.
Teaching Suggestion 8.3: Labor Planning Problem—Hong Kong Bank of Commerce.
This example is a good practice tool and lead-in for the Chase Manhattan Bank case at the end of the chapter. Without this example, the case would probably overpower most students.
Teaching Suggestion 8.4: Ingredient Blending Applications.
Three points can be made about the two blending examples in this chapter. First, both the diet and fuel blending problems presented here are tiny compared to huge real-world blending problems. But they do provide some sense of the issues to be faced.
Second, diet problems that are missing the constraints that force variety into the diet can be terribly embarrassing. It has been said that a hospital in New Orleans ended up with an LP...

...Forum #2: LinearEquations in Real Life
Pick one of the following problems. Show how you would solve it using a system
of linearequations.
1) John spent $201 shirts and pants for work. Shirts cost $27 and pants cost $22. If he
bought a total of 8 articles of clothing, then how many of each kind did he buy?
2) A school dance has 228 students. There are 63 fewer girls than twice as many boys.
How many boys and girls attended the dance?
3) There are 15 animals in the barn. Some are ducks and some are pigs. There are 46 legs in
all. How many of each animal are there?
4) The sum of two numbers is 66. The larger number is three more than twice the smaller
number. Find the numbers.
5) A police academy is training 14 new recruits. Some are working dogs and others are
police officers. There are 38 legs in all. How many of each type of recruit are there?
6) A Boy Scout troop carries 57 boys on a trip in 8 vehicles. Each van can carry 14 boys,
while each car can carry 3 boys. How many vans and how many cars did the troop take?
7) A high school purchases two workbooks for every textbook, and two journals for every
workbook. If the school purchases 301 total items, how many of each do they need?
8) A mom bought 28 packages of Reese’s Cups with 88 total cups. Each king size package
holds 4 cups. Each regular size package holds two cups. How many of each size did she
buy?
9) A high school marching band of 205 students went to a...

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