Cramer’s Rule
Introduction
Cramer’s rule is a method for solving linear simultaneous equations. It makes use of determinants and so a knowledge of these is necessary before proceeding.
1. Cramer’s Rule  two equations
If we are given a pair of simultaneous equations a1 x + b1 y = d1 a2 x + b2 y = d2 then x, and y can be found from d1 b1 d2 b2 a1 b1 a2 b2 a1 d1 a2 d2 a1 b 1 a2 b 2
x=
y=
Example Solve the equations 3x + 4y = −14 −2x − 3y = 11
Solution Using Cramer’s rule we can write the solution as the ratio of two determinants. −14 4 11 −3 3 4 −2 −3 −2 = 2, −1 3 −14 −2 11 3 4 −2 −3
x=
=
y=
=
5 = −5 −1
The solution of the simultaneous equations is then x = 2, y = −5.
5.2.1
copyright c Pearson Education Limited, 2000
2. Cramer’s rule  three equations
For the case of three equations in three unknowns: If a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3
then x, y and z can be found from d1 d2 d3 a1 a2 a3 b1 b2 b3 b1 b2 b3 c1 c2 c3 c1 c2 c3 a1 a2 a3 a1 a2 a3 d1 d2 d3 b1 b2 b3 c1 c2 c3 c1 c2 c3 a1 a2 a3 a1 a2 a3 b1 b2 b3 b1 b2 b3 d1 d2 d3 c1 c2 c3
x=
y=
z=
Exercises Use Cramer’s rule to solve the following sets of simultaneous equations. a)
...Solving systems of linearequations
7.1 Introduction
Let a system of linearequations of the following form:
a11 x1
a21 x1
a12 x2
a22 x2
ai1x1 ai 2 x2
am1 x1 am2 x2
a1n xn
a2 n x n
ain xn
amn xn
b1
b2
bi
bm
(7.1)
be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m;
k = 1, 2, ..., n) are the coefficients, bi (i = 1, 2, ..., m) are the free terms of the system. In
matrix notation, this system has the form:
Ax b ,
(7.2)
where A is the matrix of coefficients of the system (the main matrix), A = [aik]mn, b is the
column vector of the free terms, bT [b1 , b2 , ... , bm ] , x is the column vector of the
unknowns, xT [ x1 , x2 , ... , xn ] ; the symbol () T denotes transposition.
It is assumed that aik and bi are known numbers. An ordered set, {x1, x2, ..., xn}, of real
numbers satisfying (7.1) is referred to as the solution of the system, and the individual
numbers, x1, x2, ..., xn, are roots of the system.
A system of linearequations is:
consistent  if it has at least one solution. At the same time it can be

determined  if it has exactly one, unique solution,
undetermined  if it has infinitely many solutions;
inconsistent  if it does not have any solution.
The further...
...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: 2xy=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/linearequations.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 20103 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 timeandahalf 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 by using the quadratic formula: [pic]...
...Practical Applications:
Graphing SimultaneousEquations
−
−
−
−
Relating linear graphs and simultaneousequations
Analysing graphs
Practical applications of linear graphs
Writing algebraic equations
Jane Stratton
Objectives:
• Use linear graphs to solve simultaneousequations
• Use graphs of linearequations to solve a range of
problems
• Translate worded problems into graphical and algebraic
form
Finding the Solution to an Equation from a graph
• Finding solutions to an equation when we have a graph is
easy, we just need to find the coordinates of points on the
line.
𝒚 = 𝟐𝒙 − 𝟓
• Example:
𝑥 = 5 is the solution of 2𝑥 − 5 = 5
𝑥 = 4 is the solution of 2𝑥 − 5 = 3
𝑥 = 2 is the solution of 2𝑥 − 5 = −1
𝑥 = 1 is the solution of 2𝑥 − 5 = −3
𝑥 = 0 is the solution of 2𝑥 − 5 = −5
SimultaneousEquations and Graphs
• Remember: Simultaneousequations are solved at the
same time – they are two equations with the same
solutions.
• Solvingsimultaneousequations using a graph is easy when
you remember that the solution is where the 𝑥 and 𝑦
values are the same for both lines!
• This means you need to
draw the lines of both the
equations on the same graph.
• The point...
...Cramer’sRuleCramer’srule is a method of solving a system of linearequations through the use of determinants.
Matrices and Determinants
To use Cramer’sRule, some elementary knowledge of matrix algebra is required. An array of numbers, such as
6 5 a11 a12
A =
3 4 a21 a22
is called a matrix. This is a “2 by 2” matrix. However, a matrix can be of any size, defined by m rows and n columns (thus an “m by n” matrix). A “square matrix,” has the same number of rows as columns. To use Cramer’srule, the matrix must be square.
A determinant is number, calculated in the following way for a “2 by 2” matrix:
a11 a12
A = = a11 a22  a21 a12
a21 a22
For example, letting a11 = 6, a12 = 5, a21 = 3, a22 = 4:
6 5
A= = 6 (4)  3 (5) = 9
3 4
For “m by n” matrices of orders larger than 2 by 2, there is a general procedure that can be used to find the determinant. This procedure is best explained as an example. Consider the determinant for a 3 by 3 matrix
a11 a12 a13
A = a21 a22 a23...
...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 SlopeIntercept 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 PointSlope 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...
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