Transcript 14chapRules

Simultaneous Equations




Graphical Solutions
Solution by Addition or Subtraction
Solution by Substitution
Determinants
Graphical Solutions

Two linear equations plotted
on the same coordinate axes
will intersect at a single point
2x - 3y = 2
x + 2y = 8
Lines intersect at one point only:
Exactly one solution x=4, y=2
Unless:


The graphs are parallel
The graphs are co-linear
4x + 6y = 12
2x + 3y = -6
Lines are parallel:
No solution
2x – 3y = -6
-x + 3/2 y = 3
Lines coincide:
Infinite solutions
Graphical Solutions (2)
Solve:
x y 7
 
4 3 12
x y 1
 
2 4 4
Solution by Addition or
Subtraction
Put equations in the form
(1)
a1x + b1y = c1
(2)
a2x + b2y= c2
 Multiply the equations values that will produce the
same coefficient for x or y.
 Add or subtract the equations to remove one of the
variables
Solve:
7 + y = 3x – 3
5–x=2–y

Solution by Substitution
Put equations in the form
y = a1x + b1 or x = a1y + b1
 Replace one variable with its equivalent
expression
 Solve for the remaining variable
Solve
y = 0.4x
y = 50 + .3x

Determinants




A determinant is the value computed from a square matrix of
numbers by a rule of combining products of the matrix entries
a
b
c
d
= ad -bc
The number of rows (or columns) is called the order of the
determinant.
The diagonal, from the upper left corner to the lower right
corner, is called the principal diagonal.
The diagonal from the lower left corner to the upper right corner
is called the secondary diagonal.
Determinants (2)


Using Determinants to solve linear equations
Put equations in the form
(1)
a1x + b1y = k1
(2)
a2x + b2y = k2
The denominator for both
unknowns is:
a1b2 – a2b1
The numerator of x is
k1b2 – k2b1
The numerator of y is
a1k2 – a2k1