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icang [17]
3 years ago
14

Here is a linear equation in two variables: 2x+4y−31=123

Mathematics
1 answer:
Paha777 [63]3 years ago
8 0

Answer:

y=−11x+77/2

Step-by-step explanation:

The procedure for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations.[4]

Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry. In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations.

The first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule. Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy.[5]

In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb.

Linear algebra grew with ideas noted in the complex plane. For instance, two numbers w and z in {\displaystyle \mathbb {C} }\mathbb {C}  have a difference w – z, and the line segments {\displaystyle {\overline {wz}}}{\displaystyle {\overline {wz}}} and {\displaystyle {\overline {0(w-z)}}}{\displaystyle {\overline {0(w-z)}}} are of the same length and direction. The segments are equipollent. The four-dimensional system {\displaystyle \mathbb {H} }\mathbb {H}  of quaternions was started in 1843. The term vector was introduced as v = x i + y j + z k representing a point in space. The quaternion difference p – q also produces a segment equipollent to {\displaystyle {\overline {pq}}.}{\displaystyle {\overline {pq}}.} Other hypercomplex number systems also used the idea of a linear space with a basis.

Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group. The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".[5]

Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the work later.[6]

The telegraph required an explanatory system, and the 1873 publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression. Linear algebra is flat differential geometry and serves in tangent spaces to manifolds. Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations, and much of the history of linear algebra is the history of Lorentz transformations.

The first modern and more precise definition of a vector space was introduced by Peano in 1888;[5] by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations.[5]

Vector spaces

Main article: Vector space

Until the 19th century, linear algebra was introduced through systems of linear equations and matrices. In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract.

A vector space over a field F (often the field of the real numbers) is a set V equipped with two binary operations satisfying the following axioms. Elements of V are called vectors, and elements of F are called scalars. The first operation, vector addition, takes any two vectors v and w and outputs a third vector v + w. The second operation, scalar multiplication, takes any scalar a and any vector v and outputs a new vector av. The axioms that addition and scalar multiplication must satisfy are the following. (In the list below, u, v and w are arbitrary elements of V, and a and b are arbitrary scalars in the field F.)[7]

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5•10^6 is how many times as large as 5•10^6
Mamont248 [21]

Answer: 100

Step-by-step explanation:

For any two numbers X and Y, you can figure out how many times larger X is than Y by dividing X by Y . For example , 8 is 4 fine larger than 2 because 8/2=4 5*10^6/5*10^4 = 5/5*10^6/10^4 = 1*10^2 = 100

5 0
3 years ago
For the month of March in a certain city, 57% of the days are cloudy. Also in the month of March in the same city, 55% of the da
oksian1 [2.3K]

Answer: 0.9649

Step-by-step explanation:

Let A denote the event that the days are cloudy and B denotes the event that the days are rainy.

Given : For the month of March in a certain city, the probability that days are cloudy :P(A)=0.57

Also in the month of March in the same city,, the probability that the days are cloudy and rainy :P(A\cap B)=0.55

Now by using the conditional probability, the probability that a randomly selected day in March will be rainy if it is cloudy will be :-

P(B|A)=\dfrac{P(A\cap B)}{P(A)}

\Rightarrow\ P(B|A)=\dfrac{0.55}{0.57}\\\\=0.964912280702\approx0.9649\ \ \text{[Rounded to four decimal places.]}

Hence, the probability that a randomly selected day in March will be rainy if it is cloudy = 0.9649

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Solve similar triangles can someone please answer
Vilka [71]

Answer:

Step-by-step explanation:

\frac{XY}{PQ}=\frac{YZ}{QR}\\\\\frac{19.5}{6.5}=\frac{10.5}{k}\\\\

Cross multiply,

19.5*k = 10.5 * 6.5

k =\frac{10.5*6.5}{19.5}\\\\k=3.5

8 0
3 years ago
HEY PLEASE I NEED HELP IM OFFERING 30 POINTS
Levart [38]

Answer:

the first option, (-4, 6)

Step-by-step explanation:

for this question, we can use the substitution method. this is where we substitute the coordinates in the answer choices into the given system of inequalities or equations to determine which one is true.

1. (-4,6)

x + 4y > 12

-4 + 4(6) > 12

20 > 12 is true

now we can substitute the second inequality.

3(6) > -4 + 6

18 > 2 is true, therefore making this set of points true.

it may be helpful to substitute the rest of the values of x and y into the inequalities to verify your answers :

5 + 4(2) > 12

5 + 8 > 12

13> 12 is true

3(2) > 5 + 6

6 > 11 is not true

the rest of the answer choices are also not true.

I hope this helped you!

8 0
3 years ago
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