Download Systems of Equations

Algebra 2 – Unit 13 – Systems of Equations - NEED TO KNOW…..
A. Terminology –
a. System of Linear Equations is 2 or more equations considered together.
b. Solution of a System of Linear Equations is the ordered pair ( x, y ) (if it
exists) that when substituted into each linear equation in the system of
linear equations makes each TRUE.
c. Independent System of Linear Equations – When a system of equations
has only one solution it is called Independent. The graph of the system
will show equations intersecting at only one point.
d. Inconsistent System of Linear Equations – When a system of equations
has no intersecting points then there is no solution and it is called
Inconsistent. The graph of the system will show parallel lines.
e. Dependent System of Linear Equations – When all solutions for one
equation work for all the other equations in the system the system is
called Dependent. The graph of the system will be the same lines. The
solution looks like the ordered pair, ( x, y solved in terms of x) .
f. Matrix – A rectangular array of numbers. Each number in the matrix is
called an element. Each element is denoted by its row and column.
 a11 a12 
a
 This is called a 2 x 2 matrix or a square matrix.
 21 a22 
g. Determinant – A number that is associated with a matrix. It is a
a
a
formula: 11 12  (a11 )(a22 )  (a21 )(a12 )
a21 a22
h. Cramer’s Rule – Method to solve a System of Linear Equations. If you
have:
a11 x  a12 y  b1
a21 x  a22 y  b2 Then the solution (x, y ) is found by ( x 
D=
a11
a12
a21
a22
, Dx 
b1
a12
b2
a22
, Dy 
a11
b1
a21 b2
Dy
Dx
,y
) where
D
D
, & D  0.
i. Systems of Linear Inequalities are solved Graphically by shading in the
region of intersection if one exists once the inequalities are graphed.
B. Methods to Solve Systems of Linear Equations –
a. Graphing Method - Just graph the equations and find the point of
intersection, if one exists. The solution is the point of intersection, an
ordered pair ( x, y ) .
b. Substitution Method – Solve one of the equations in the system for one
of the variables. Then substitute into the other equation for that variable
and solve. Take that variable solution and solve to find the value of the
other variable.
c. Addition Method – The goal using this method is to get rid of one of the
variables by adding the equations together. To do this you may have to
multiply one or both of the equations by something so when you add the
equations together one of the variables will cancel out. Once a variable
is cancelled out then use Substitution to finish the problem.
d. Cramer’s Rule - Method to solve a System of Linear Equations. If you
have a system of linear equations that looks like:
a11 x  a12 y  b1
a21 x  a22 y  b2 Then the solution (x, y ) is found by ( x 
D=
a11
a12
a21
a22
, Dx 
b1
a12
b2
a22
, Dy 
a11
b1
a21 b2
Dy
Dx
,y
) where
D
D
, & D  0.
C. Method to Solve Systems of Inequalities - Systems of Linear Inequalities are solved
Graphically by shading in the region of intersection, if one exists, once the
inequalities are graphed. The following websites demonstrate how to
graph and solve systems of inequalities:
http://www.purplemath.com/modules/syslneq.htm - This
website gives good examples.
http://www.mathwarehouse.com/algebra/linear_equation/systems-ofequation/system-linear-inequality.php - This website is interactive.