Systems of two simultaneous linear equations in two unknowns
Systems of two simultaneous linear equations in two unknowns.
Basic methods of solution. Substitution. Addition or subtraction
of equations. The second order determinants. Cramer's rule.
Investigation of solutions.
Systems
of two simultaneous linear equations in two unknowns have the shape:
where a, b, c, d, e, f – numerical coefficients; x,
y – unknowns.
Solution
of these simultaneous equations can be found by two basic
methods:
Substitution. 1).
From one equation we express one of unknowns, for example x, by coefficients and another unknown y :
x = ( c – by
) / a , (2)
2). Substitute in the second equation instead of x :
d ( c –
by ) / a + ey = f .
3).
Now, solving the last equation, find y :
y
= ( af – cd ) / ( ae – bd ).
4). Substitute this value for y in the expression (2) instead of y :
x = ( ce
– bf ) / ( ae – bd ) .
E x a m p l e . Solve
the system of simultaneous equations:
From the first equation express x by coefficients
and y :
x =
( 2y + 4 ) / 3 .
Substitute
this expression into the second equation and find y :
( 2y +
4 ) / 3 + 3y = 5 , hence y = 1 .
Now find x , substituting the found value instead
of y into
expression for x: x = ( 2 · 1 + 4 ) / 3 , from
here x = 2 .
Addition
or subtraction. This
method consists in the following.
1).
Multiply both sides of the first equation of the system (1) by ( – d ) and both sides
of
the second equation by a and add them:
From here we receive: y = ( af – cd ) / ( ae – bd ) .
2).
Substitute the found value of y into any equation of the original system (1)
:
ax + b( af – cd ) / ( ae – bd ) = c .
3).
Find another unknown x : x = ( ce – bf ) / ( ae – bd ) .
E
x a m p l e . Solve the system of simultaneous equations:
by the second way ( addition or subtraction ).
Multiply the first equation by –1, the second by 3 and
add them:
From here y = 1 .
Substitute this value into the second equation
( is it possible to
substitute this into the first equation ? ): 3x +
9 = 15, hence, x = 2 .
The
second order determinants. We
saw, that formulas for solution of the
system
of two simultaneous linear equations in two unknowns have
the shape:
x = ( ce
– bf ) / ( ae – bd ) ,
(3)
y = ( af – cd ) / ( ae – bd ) .
These
formulas can be remembered very easily, if to introduce for
their numerators
and denominators the next symbol:
, which will be used to mean an expression: ps
– qr .
This
expression is received by crosswise multiplication of numbers p, q, r, s :
and
the following subtraction of one product from another: ps
– qr. The sign “+”
is taken for a product of numbers, located on the diagonal,
going from the left
upper number to the right lower number. The sign “– “
for another diagonal, going
from the right upper number to the left lower number. For example,
The expression is called the second order determinant.
Cramer’s
rule. Using
the determinants, the formulas (3) can be written as:
Formulas
( 4 ) are called Cramer’s rule for solution of the
system of two simultaneous
linear equations in two unknowns.
E
x a m p l e . Solve the system of simultaneous equations
using Cramer’s rule.
S
o l u t i o n . Here a = 1, b = 1, c =
12, d = 2, e = –3, f = 14 .
Investigation
of solutions of
a system of two simultaneous linear equations in two unknowns
shows, that depending on coefficients three different
cases are possible:
1) coefficients
at unknowns in equations are disproportionate:
a : d ≠ b : e ,
in this case the system of simultaneous linear equations
has a single solution,
presented by formulas (4) ;
2) all
coefficients of equations are proportional: a:
d = b: e = c: f , in this case
the system of simultaneous linear equations has an infinite
set of solutions,
because we have actually one equation instead of two.
E x a m p l e . In the system
and
this system has an infinite set of solutions. ( Why? )
Dividing
the first equation by 2 and the second  by 3,
we’ll receive two identical equations:
that is one equation in two unknowns, which has an infinite
set of solutions.
3) coefficients
at unknowns are proportional, but disproportionate to free
terms:
a : d = b
: e ≠ c : f , in this case the system of simultaneous linear equations
has
no
solutions, because we have here contradictory equations.
terms is 7/12, not equal
to 1/3. Why has not this system solution?
An answer is very easy. If to divide the second equation
by 3,
we’ll receive:
These
equations are contradictory, because the same expression
2x – 3y cannot be equal both to 7 and 4
simultaneously.
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