- Source: Indeterminate equation
In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation
a
x
+
b
y
=
c
{\displaystyle ax+by=c}
is a simple indeterminate equation, as is
x
2
=
1
{\displaystyle x^{2}=1}
. Indeterminate equations cannot be solved uniquely. In fact, in some cases it might even have infinitely many solutions. Some of the prominent examples of indeterminate equations include:
Univariate polynomial equation:
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
2
x
2
+
a
1
x
+
a
0
=
0
,
{\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=0,}
which has multiple solutions for the variable
x
{\displaystyle x}
in the complex plane—unless it can be rewritten in the form
a
n
(
x
−
b
)
n
=
0
{\displaystyle a_{n}(x-b)^{n}=0}
.
Non-degenerate conic equation:
A
x
2
+
B
x
y
+
C
y
2
+
D
x
+
E
y
+
F
=
0
,
{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,}
where at least one of the given parameters
A
{\displaystyle A}
,
B
{\displaystyle B}
, and
C
{\displaystyle C}
is non-zero, and
x
{\displaystyle x}
and
y
{\displaystyle y}
are real variables.
Pell's equation:
x
2
−
P
y
2
=
1
,
{\displaystyle \ x^{2}-Py^{2}=1,}
where
P
{\displaystyle P}
is a given integer that is not a square number, and in which the variables
x
{\displaystyle x}
and
y
{\displaystyle y}
are required to be integers.
The equation of Pythagorean triples:
x
2
+
y
2
=
z
2
,
{\displaystyle x^{2}+y^{2}=z^{2},}
in which the variables
x
{\displaystyle x}
,
y
{\displaystyle y}
, and
z
{\displaystyle z}
are required to be positive integers.
The equation of the Fermat–Catalan conjecture:
a
m
+
b
n
=
c
k
,
{\displaystyle a^{m}+b^{n}=c^{k},}
in which the variables
a
{\displaystyle a}
,
b
{\displaystyle b}
,
c
{\displaystyle c}
are required to be coprime positive integers, and the variables
m
{\displaystyle m}
,
n
{\displaystyle n}
, and
k
{\displaystyle k}
are required to be positive integers satisfying the following equation:
1
m
+
1
n
+
1
k
<
1.
{\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}<1.}
See also
Indeterminate form
Indeterminate system
Indeterminate (variable)
Linear algebra
References
Kata Kunci Pencarian:
- Indeterminate equation
- Indeterminate
- Diophantine equation
- Statically indeterminate
- Polynomial
- Equation
- Indeterminate form
- Pell's equation
- Indeterminate (variable)
- Quadratic equation
