Solving System of Equations

In mathematics, a system of linear equations is a collection of linear equations involving the same set of variables. For example,

is a system of three equations in the three variables . A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by

since it makes all three equations valid.so this is specially for linear equations in two variables or more than two variables.

In mathematics, the theory of linear systems is a branch of linear algebra, a subject which is fundamental to modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and such methods play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.This is how linear equations explained ,let's see a example on this now from linear algebra tutoring online.
Question:-

Solve the system of equations

8x-z=4

y+z=5

11x+y=15

Answer:-

8x-z=4 ----- eq1

y+z=5 ---- eq2

11x+y=15 -----eq3

add eq1 and eq2

we get 8x+y=9 ----- eq4

subtract eq4 from eq3

11x+y=15
-8x+y=9
-----------
3x = 6

x =2

putting this value in eq1

8x-z =4

16-z=4

z=12

putting this value in eq2

y+z = 5

y = -7

so x=2 , y= -7 and z=12

How to find the arc length of a curve

The arc length is a topic from 10 grade math geometry,lets see the general definition for arc length after which we can see what is the arc length formula and how to use it.
A curve in, say, the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment (using the theorem of Pythagoras in Euclidean space, for example), the total length of the approximation can be found by summing the lengths of each linear segment.

This free online tutoring math will discuss this in more detail like,
If the curve is not already a polygonal path, better approximations to the curve can be obtained by following the shape of the curve increasingly more closely. The approach is to use an increasingly larger number of segments of smaller lengths. The lengths of the successive approximations do not decrease and will eventually keep increasing—possibly indefinitely, but for smooth curves this will tend to a limit as the lengths of the line segments get arbitrarily small.

For some curves there is a smallest number L that is an upper bound on the length of any polygonal approximation. If such a number exists, then the curve is said to be rectifiable and the curve is defined to have arc length

The arc length formula is s=rθ

where s is the arc , r is the radius of the circle and θ is the angle .

Commutative and Distributive Number Properties

Number properties are
Associative property applies to both addition and multiplication: a+(b+c)=(a+b)+c, a*(b*c)=(a*b)*c.
Distributive property ties addition and multiplication together. Distributive property can be written as: a*(b+c) = a*b + a*c.
Commutative property means that operands can switch places: a+b=b+a, a*b=b*a.

For Example:
Question 1 :- Distributive PropertyThe property demonstrated in the equation -(3n - 5) = -3n + 5 is:
a. associative b. commutative c. distributive d. Neither side of the equation is equal to the other.
Answer:
-1(3n-5) = -3n+5
(-1) (3n) + (-1) (-5)
= -3n+5 (Distributive Property)

Question 2 :- Associative Property
property demonstrated in the equation (5 + n) - 2 = 5 + (n - 2) is:
Answer:
(5+n)-2 = 5+(n-2)
Here the grouping that is the parentheses has changed.so it a (Associative Property)

Question 3 :- Commutative Property
property is this one 7 - 4 + 2 = 7 + 2 – 4
Answer:
7 - 4 + 2 . 7 + 2 - 4
Here the order - 4 + 2 has become 2 + - 4. So it is a (Commutative Property)