This document discusses linear systems of equations. It defines a linear system as one where the equations are of the first degree. It describes four methods for solving linear systems: substitution, comparison, reduction, and Cramer's method. It also discusses literal systems that contain parameters and fractional systems where variables appear in denominators. An example demonstrates solving a 3x3 system using substitution to find the unique solution.

1. Linear
Systems
School year 2014/2015

2. Systems
A system consists of a set of equations, for
which we ask what are the common solutions.
ax+by=c
a’x+b’y=c’

3. System Degree
It’s the product of the higher degrees of the
individual equations that make up the system.
EXAMPLE:
4x-y²=3
x5-y=10
2 * 5 = gr.10

4. So a linear system is of first degree.

5. How do you solve linear systems?
- Substitution method
- Comparison method
- Reduction method
- Cramer’s method

6. Substitution method

7. Derive a variable by an equation and replace it
in the other equation.
ESEMPIO:
2x-5y=7
x-3y=1
x=3y+1
2(3y+1)-5y=7
x=3y+1
6y+2-5y=7
x=16
y=5

8. Comparison method

9. Derive the same variable
from both equations
a’x + b’y = c’
canonical
form
ax + by = c

10. x - y = 3
x + y = 9
x = y + 3
x = -y + 9
x = y + 3
y + 3 = -y + 9
x = y + 3
2y = 6
x = 6
y = 3
Example:

11. Reduction method

12. Reduction Method
It’s to add or subtract
the corresponding terms of the two equations
to obtain an equation with only one unknow.

13. EXAMPLE: PROCEDURE:
1. Multiplie one or both of the equations for factors
non-zero, so that the coefficients of one of the
variables are equal to or opposite.
2. If the coefficients obtained in step 1 are equal,
subtract member to member the two equations; if
the coefficients are opposite, add member to
member; so we get an equation in one unknown.
3. Solve the equation in a single variable.
4. Replace the solution in one of the two original
equations.

14. Cramer’s method

15. The system must be written in canonical form:
ax+by=c
a'x+b' y=c'

16. Determinant calculation (D)
+
-
= +ab'-a'b
Multiply +a with b' and -a' with b
a b
D=
a' b'

17. Dx calculation
+
-
= +cb'-bc'
Multiply +c with b' and -c' with b
c b
Dx=
c' b'

18. -
Dy calculation
+
= +ac'-a'c
Multiply +a with c' and -a' with c
a c
Dy=
a' c'

19. If D ≠ 0 the
system is
determined
x=Dx/D
y=Dy/D
If D = 0
-the system is
indeterminate with
Dx and Dy=0
-the system is
impossible with Dx
and Dy≠0

20. Literal Systems

21. The literal systems are those where in
addition to the variables there are other
letters (parameters).

22. Example
Transform the system in canonical form.
2x= 2a-y 2x+y=2a
(a+1)x+ay=2a (a+1)x+ay=2a
For literal system the most used method is Cramer, calculating the determinant D, Dx, Dy.
D= = 2a-(a+1) = 2a-a-1= a-1 Dx= = 2a2 -2a=2a(a-1)
Dy= = 4a -2a(a+1)=4a-2a2 -2a=2a-2a2 =2a(1-a)
2 1
a+1 a
2a 1
2a a
2 2a
a+1 2a

23. ...continuous example
The system is determined if D ≠ 0 ie if a-1≠0 a≠1.
● If a≠1 then
● If a=1 then D=0, Dx=0 and Dy=0 and the system is indeterminated.
x= Dx/D= 2a(a-1)/a-1= 2a
y=Dy/D= 2a(a-1)/a-1= -2a(a-1)/a-1= -2a
x= 2a
y= -2a

24. Linear Fractional Systems

25. When a system is fractional?
● Are those systems in which at least one of the equations that compose
it appears the unknown of first degree (x; y) in the denominator.
● Is solved with the methods we have already seen. (eg. the
replacement method; method of comparison; reduction method etc ...),
but it should be the
EXISTENCE CONDITION (E. C.)
steps shall be non-zero
all denominators that contain the unknown

26. Example

27. Found:
● l.c.m= 2xy
● E.C.: x≠0 U y ≠0

28. Transform the system in canonical form...
…and choose the most appropriate method to solve it.

30. Result:

31. Check if the solution of the system
satisfies the E.C.

32. Sistems with 3 equations and 3 variables

33. CANONICAL FORM

34. Solve operations in brackets

35. Order and simplify the terms like putting the system in CANONICAL FORM

36. CANONICAL FORM

37. Find the value of y will go out and replaced in the other two equations using the
method of substitution
So the coefficient of y of the first equation is equal to 1, derive the value of y

38. Solve in order to remove the brackets

39. Order and simplify the similar terms in the equations in which the value replaced

40. The first equation must be simplified for 5

41. Since the coefficient of z of the first equation is equal to 1, derive from it the
value of z

42. Found the value of z and replace in the other two equations using
the substitution method

43. Solve operations in brackets

44. Order and simplify opposite terms

45. Find the value of x in the first equation and substitute in last

46. The system solution is given by the triplet (4; 0 ; 5 ) that simultaneously
solves all of the system equations. The system is therefore DETERMINED .

47. WORK MADE BY THE CLASS 2nd A Afm
OF ITCG “CORINALDESI” - SENIGALLIA (AN) - ITALY
Team 1 - Breccia Martina, Franceschetti Sofia, Pinca Julia Andrea
Team 2 - Valentini Alessia, Esposto Giorgia, Biagetti Elena, Montironi Ilaria, Carletti
Lucia
Team 3 - Fabri Luca, Franceschini Simone, Bernardini Alessio, Urbinelli Riccardo,
Saramuzzi Mirko
Team 4 - Rossi Davide, Ventura Devid, Latini Angelo
Team 5 - Cervasi Michela, Casella Federica, Raccuja Ilaria
Team 6 - Zhang Qiuye , Zhang Ting, Xie Sandro
Team 7 - Trionfetti Sara, Carbonari Gloria, Borgacci Francesca, Avaltroni Alessia