30 20 10 -10 -20 -10 Linear equations in n variables: a,x, + a,x, + a,x, ++ a + a,x, = b (a,, a, a a,: constants) Solutions: Graph: b-a,x, -a,x, a X,, X3, X X Hyperplane in n dimensions a. Systems of Linear Equations Case Li (unique solution) * + 2y = 3 4x + 9y =8 10
30 20 10 -10 -20 -10 Linear equations in n variables: a,x, + a,x, + a,x, ++ a + a,x, = b (a,, a, a a,: constants) Solutions: Graph: b-a,x, -a,x, a X,, X3, X X Hyperplane in n dimensions a. Systems of Linear Equations Case Li (unique solution) * + 2y = 3 4x + 9y =8 10
30 20 10 -10 -20 -10 Linear equations in n variables: a,x, + a,x, + a,x, ++ a + a,x, = b (a,, a, a a,: constants) Solutions: Graph: b-a,x, -a,x, a X,, X3, X X Hyperplane in n dimensions a. Systems of Linear Equations Case Li (unique solution) * + 2y = 3 4x + 9y =8 10
I need help understanding how to use the n variables formula and the 3 equations formula in my professors pdf. Basically I need more claification on how to use these formulas. Please repsond soon.
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Linear equations in n variables:
a,X, + a,x2 +azx3 + • ··+ a,-Xn-1 + a „x, = b
(а,, а>, аз»
constants)
Solutions:
Graph:
b-a,x - a,X2
a
n-1n-1
X1, X2, X3, •…., Xn-12
Hyperplane in n dimensions
an
Systems of Linear Equations
Case I: (unique solution)
x + 2y = 3
4х + 9у %3D 8
4
||
-10
10
15
5
-2-
-4
II
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Linear Equations and Systems
Linear equations in one variable:
1
ax = b
(a,b: constants)
Solution:
x =
(a ±0)
Linear equations in two variables:
ах + by %3D с
(a,b,c: constants)
Solutions:
Graph:
c-by
С — ах
х,
b
or
Line in 2 dimensions
а
1
- 1
4
Linear equations in three variables:
ах + by + cz %3D d
(a,b,c,d: constants)
Solutions:
Graph:
d - ах -by
or
d — ах — Cz
х,
х, у,
Plane in 3 dimensions
or ...
5
b
II
Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.
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