**Patterns Within Systems Of Linear Equations Mathematics Essay**

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The intent of this study is to look into systems of additive equations where the systems ‘ invariables have mathematical forms.

The first system to be considered is a 2 ten 2 system of linear of equations:

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In the first equation, the invariables are 1, 2, and 3 in that order. It is observed that each invariable is increased by 1 from the old invariable. Therefore, the invariables make up an arithmetic sequence whereby the first term ( U1 ) = 1, and the difference between each term ( vitamin D ) = 1. Hence, the general expression is Un = U1 + ( n-1 ) ( 1 ) where n represents the n-th term.

U2 = U1 + ( n-1 ) ( vitamin D ) U3 = U1 + ( n-1 ) ( vitamin D )

2 = 1 + ( 2-1 ) ( 1 ) 3 = 1 + ( 3-1 ) ( 1 )

2 = 2 3 = 3

In the 2nd equation, the invariables are 2, -1, and -4 in that order. It is observed that each invariable is increased by -3 from the old invariable. Therefore, the invariables from this equation besides make up an arithmetic sequence whereby U1 = 2 and d = -3. Hence the general expression is Un = U1 + ( n-1 ) ( -3 ) .

U2 = U1 + ( n-1 ) ( vitamin D ) U3 = U1 + ( n-1 ) ( vitamin D )

-1 = 2 + ( 2-1 ) ( -3 ) -4 = 2 + ( 3-1 ) ( -3 )

-1 = -1 -4 = -4

To further look into the significance of these arithmetic sequences, the equations will be solved by permutation and displayed diagrammatically.

ten + 2y = 3 2x – Y = -4

ten = 3- 2y Y = 2x + 4

ten = 3 – 2 ( 2 ) Y = 2 ( 3 – 2y ) +4

ten = -1 5y = 6 + 4

Y = 2

On the graph, both lines meet at a common point ( -1,2 ) where ten = -1 and y = 2.

The two additive equations have a solution of x = -1 and y = 2, proven analytically and diagrammatically.

However, this form may be merely specific to this 2 x 2 system of additive equations. Therefore, other 2 x 2 system of additive equations following the same form of holding invariables organizing arithmetic sequences will be examined every bit good.

Another 2 x 2 system of additive equations to be considered is:

The invariables of these equations are 3, 6, and 9, and 4, 2, and 0 with a difference of 1 and -2 severally.

The equations were so re-written as:

And plotted on a graph.

The common point of both equations is ( -1,2 ) , with ten being -1 and y being 2. Therefore the common point has been proven both analytically and diagrammatically to be ( -1,2 ) .

Another illustration is:

The invariables of these equations are -3, 1, and 5, and -2, -6, and -10 with a difference of 4 and -4 severally.

The equations were so re-written as:

And plotted on a graph.

The common point is ( -1,2 ) . Thus it is both proved analytically and diagrammatically that the common point is ( -1,2 ) .

Another illustration is:

The invariables of these equations are 3, 2, and 1, and 2, 7, and 12 with a difference of -1 and 5 severally.

The equations were so re-written as:

And plotted on a graph.

The common point is ( -1,2 ) . Thus it is both proved analytically and diagrammatically that the common point is ( -1,2 ) .

Another illustration is:

The invariables of these equations are 5, 12, and 19, and 1, -5, and -11 with a difference of 7 and -6 severally.

The equations were so re-written as:

And plotted on a graph.

The common point is ( -1,2 ) . Thus it is both proved analytically and diagrammatically that the common point is ( -1,2 ) .

From the illustrations of 2×2 systems of additive equations, a speculation that could be derived is:

“ The solution for any 2×2 system of additive equations with invariables that form an arithmetic sequence is ever x=-1 and y=2. ”

The general expression of such equations could be written as:

Whereby represents the first term for the first equation and represents the first term for the 2nd equation with a common difference of and severally.

The equations are so solved at the same time:

Therefore, it is proved that the solution for a 2×2 system of additive equations with invariables that form an arithmetic sequence is ever ten = -1 and y = 2.

However, the possibility of a 3×3 system exhibiting the same forms as the old 2×2 systems examined has non been discussed. Hence, this probe will widen to 3×3 systems every bit good.

Here is an 3×3 system:

The for the first equation is 3 and the is ( 5-3 ) = 2.

The for the first equation is 1 and the is ( -4-1 ) =-5.

The for the first equation is 4 and the is ( 7-4 ) =3.

Gaussian Elimination method will be used.

Change R3 into 4R2-R3

Change R2 into 3R2-R1

Change R3 into 23R2-17R3

The 3rd row/R3 has all 0 which means that there is no 1 alone solution but infinite solutions. Therefore, in R2

We will allow where K is a parametric quantity

To happen other solutions, will be substituted in the other equation

The solutions to this 3×3 system of additive equations with the form of invariables doing up an arithmetic sequence are, , and where is a parametric quantity.

Here is another 3×3 system:

The for the first equation is 2 and the is ( 3-2 ) = 1.

The for the first equation is 5 and the is ( 5-3 ) =-2.

The for the first equation is -3 and the is ( 4- ( -3 ) ) =7.

The equations were put into matrix signifier and row decrease was done on the Graphic Design Calculator.

The 3rd row is all 0. This indicates that there is no alone solution, but infinite solutions alternatively.

Assuming that whereby is a parametric quantity,

The solutions to this 3×3 system of additive equations with the form of invariables doing up an arithmetic sequence are, , and where is a parametric quantity.

Another:

The for the first equation is 4 and the is ( -2-4 ) = -6.

The for the first equation is 1 and the is ( 5-1 ) =-4.

The for the first equation is 2 and the is ( 7-2 ) =5.

The equations were put into matrix signifier and row decrease was done on the Graphic Design Calculator.

The 3rd row is all 0. This indicates that there is no alone solution, but infinite solutions alternatively.

Assuming that whereby is a parametric quantity,

The solutions to this 3×3 system of additive equations with the form of invariables doing up an arithmetic sequence are, , and where is a parametric quantity.

Here is another 3×3 system:

The for the first equation is 4 and the is ( -4-4 ) = -8.

The for the first equation is 2 and the is ( -1-2 ) =-3.

The for the first equation is 6 and the is ( 14-6 ) =8.

The equations were put into matrix signifier and row decrease was done on the Graphic Design Calculator.

The 3rd row is all 0. This indicates that there is no alone solution, but infinite solutions alternatively.

Assuming that whereby is a parametric quantity,

The for the first equation is 7 and the is ( 20-7 ) = 13.

The for the first equation is 20 and the is ( 3-20 ) =-17.

The for the first equation is 6 and the is ( -5-6 ) = -11.

Assuming that whereby is a parametric quantity,

From these illustrations, a speculation can be made. A 3×3 system of equations that have invariables that form an arithmetic sequence will hold infinite solutions that will be in the signifier of, , and where is a parametric quantity.

This is proven by the general expression:

Bing solved by utilizing Gaussian riddance regulation:

Change R3 into R3-R2

Change R2 into R2-R1

Change R3 into

Change R2 into

Change R3 into R3-R2

R3 has merely zeroes/0. This means that there is no alone solution but infinite solutions alternatively.

Assume whereby is a parametric quantity,

Through permutation,

The solutions for this 3×3 system are, , and, turn outing the speculation true.

Other than systems of additive equations that contain arithmetic sequences, other types will be investigated.

Let ‘s see this 2 x 2 system:

In the first equation, the invariables 1, 2, and 4 make up a geometric sequence whereby the first term ( U1 ) is 1 and each back-to-back term is multiplied by a common ratio ( R ) which in this instance is 2.

a†’ U2 a†’ U3

In the 2nd equation, the invariables 5, -1, and do up a geometric sequence whereby U1 = 5 and r = – .

a†’ U2 = a†’ U3

The equations can be rewritten in the signifier of as:

For the first equation, and.

For the 2nd equation, and..

The relationship between and appears to be that one is the negative reciprocal of the other. In any instance, more illustrations of similar additive equations will be needed to thoroughly look into the forms.

The equations will be solved by permutation:

Another illustration:

In the first equation, the invariables 3, 12, and 48 make up a geometric sequence whereby the first term ( U1 ) is 3 and each back-to-back term is multiplied by R which in this instance is 4.

a†’ U2 a†’ U3

In the 2nd equation, the invariables 3, -1, and do up a geometric sequence whereby U1 = 3 and r = – .

a†’ U2 = a†’ U3

The equations can be rewritten in the signifier of as:

For the first equation, and.

For the 2nd equation, and..

The equations will be solved utilizing permutation:

Another illustration:

In the first equation, the invariables 7, 42, and 252 make up a geometric sequence whereby the first term ( U1 ) is 7 and each back-to-back term is multiplied by R which in this instance is 6.

a†’ U2 a†’ U3

In the 2nd equation, the invariables 2, -1, and do up a geometric sequence whereby U1 = 2 and r = – .

a†’ U2 = a†’ U3

The equations can be rewritten in the signifier of as:

For the first equation, and.

For the 2nd equation, and..

The equations will be solved by utilizing permutation:

From detecting all three systems, it is found that the relationship between and appears to be that one is the negative reciprocal of the other. But it can besides be said that.

The general expression of such equations could be written as:

Whereby represents the first term for the first equation and represents the first term for the 2nd equation with a common ratio of and severally.

The equations are so solved at the same time:

So is the consequence of one ratio subtracted from the other.

is the merchandise of the common ratios from both additive equations.

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