Linear equation

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Dalam matematika, persamaan linear adalah persamaan yang dapat dimasukkan ke dalam bentuk

${\ displaystyle a_ {1} x_ {1} \ cdots a_ {n} x_ {n} c = 0,}  ÂÂ$

di mana ${\ displaystyle x_ {1}, \ ldots, x_ {n}}  ÂÂ$ adalah variabel atau tidak diketahui, dan ${\ displaystyle c, a_ {1}, \ ldots, a_ {n}}  ÂÂ$ adalah koefisien, yang sering berupa bilangan real, tetapi bisa berupa parameter, atau bahkan ekspresi apa pun yang tidak mengandung hal yang tidak diketahui. Dengan kata lain, persamaan linear diperoleh dengan menyamakan nol polinomial linier.

The solution of such equations is the values ​​which, when replaced with the unknown, make the equation true.

Kasus satu yang tidak diketahui adalah sangat penting, dan sering bahwa persamaan linear merujuk secara implisit pada kasus khusus ini, yaitu persamaan yang dapat ditulis dalam bentuk

${\ displaystyle ax b = 0.}  ÂÂ$

Jika a ? 0 persamaan linear ini memiliki solusi unik

${\ displaystyle x = - {\ frac {b} {a}}}  ÂÂ$

The solution of the linear equations in the two variables forms the line in the Euclidean plane, and each line can be defined as the solution of a linear equation. This is the origin of the term linear for qualifying this type of equation. More generally, the solution of the linear equations in the n variable forms the hyperplane (from dimension n - 1 ) in the Euclidean dimension n .

Linear equations often occur in all mathematics and their applications in physics and engineering, in part because nonlinear systems are often approached well by linear equations.

This article considers the case of a single equation with a real coefficient, in which one studies the actual solution. All of its contents apply to complex solutions and, more commonly to linear equations with coefficients and solutions in any field. For the case of some simultaneous linear equations, see System of linear equations.

Video Linear equation

Satu variabel

Persamaan linear dalam satu tidak diketahui x selalu dapat ditulis ulang

${\ displaystyle ax = b.}  ÂÂ$

Jika a ? 0 , ada solusi unik

${\ displaystyle x = {\ frac {b} {a}}.}  ÂÂ$

If a = 0 , then, if b = 0 , each number is the solution of the equation, and, if b ? 0 , there is no solution (and the equation is said to be inconsistent).

Maps Linear equation

Two variables

Persamaan linear umum dalam dua variabel x dan y adalah relasi yang menghubungkan argumen dan nilai dari suatu fungsi linear:

${\ displaystyle y = mx y_ {0},}  ÂÂ$

di mana m dan ${\ displaystyle y_ {0}}  ÂÂ$ adalah bilangan real. Grafik dari fungsi linear demikian adalah himpunan solusi dari persamaan linier ini, yang merupakan garis dalam bidang Euclidean dari kemiringan m dan y -intercept ${\ displaystyle y_ {0}}  ÂÂ$ .

Setiap persamaan linear dalam x dan y dapat ditulis ulang

${\ displaystyle ax c = 0,}  ÂÂ$

where a and b are not both zero. Solution series form a line in the Euclidean field, which is a linear function graph if and only if b ? 0 .

Using the basic algebra law, the linear equations in the two variables can be rewritten in some of the standard forms described below, often referred to as "equations of lines". In the following, x , y , t , and ? is variable; other letters represent constants (fixed numbers).

General (or standard) form

Dalam bentuk umum (atau standar) persamaan linear ditulis sebagai:

${\ displaystyle Ax = C, \,}  ÂÂ$

where A and B are not both equal to zero. This equation is usually written so that A > = 0, by convention. The equation graph is a straight line, and each straight line can be represented by the equation in the above form. If A is zero, then x accept, i.e. x -the coordinate of the point where the graph intersects x < axis (where y is zero), is C / A . If B is zero, then y -including, it is the y -the coordinate of the point where the graph intersects y - the axis (where x is zero), is C / B , and the slope of the line - A / B . Typical forms are sometimes written as:

${\ displaystyle ax c = 0, \,}$

di mana a dan b tidak keduanya sama dengan nol. Kedua versi dapat dikonversi dari satu ke yang lain dengan menggerakkan istilah konstan ke sisi lain dari tanda yang sama.

Slope-intercept form

${\ displaystyle y = mx b,}  ÂÂ$

where m is the slope of the line and b is the interference y , which is the coordinate y of the location where the line is cut axis y . This can be seen by letting x = 0, which immediately gives y = b . It may be useful to think about this in terms of y = b mx ; where the line passes through the point (0, b ) and extends to the left and right on the slope m . Vertical lines, having undefined slope, can not be represented by this form.

Bentuk yang sesuai ada untuk x intercept, meskipun itu kurang digunakan, karena y secara konvensional adalah fungsi x :

${\ displaystyle x = ny a.}  ÂÂ$

Secara analog, garis horizontal tidak dapat direpresentasikan dalam bentuk ini. Jika sebuah garis tidak horizontal atau vertikal, garis dapat dinyatakan dalam kedua bentuk ini, dengan ${\ displaystyle m \ cdot n = 1}  ÂÂ$ , jadi ${\ displaystyle m = 1/n}  ÂÂ$ . Mengekspresikan y sebagai fungsi x memberikan formulir:

${\ displaystyle y = m (x-a),}  ÂÂ$

yang setara dengan faktorisasi polinomial dari bentuk interupsi y . Ini berguna ketika intercept x lebih menarik daripada y intercept. Memperluas kedua formulir menunjukkan bahwa ${\ displaystyle b = -ma}  ÂÂ$ , jadi ${\ displaystyle a = -b/m}  ÂÂ$ , mengekspresikan x intercept dalam hal y intercept and slope, atau sebaliknya.

Bentuk titik-kemiringan

${\ displaystyle y-y_ {1} = m (x-x_ {1}), \,}  ÂÂ$

where m is the slope of the line and ( x ₁ , y ₁ ) exists point on the phone.

The point-slope form reveals the fact that the differences in y coordinate between two points on the line (i.e. /sub>) is comparable to the difference in coordinates x (ie, x Ã, - x ₁ ). The proportionality constant is m (the slope of the line).

Two-point form

${\ displaystyle y-y_ {1} = {\ frac {y_ {2} -y_ {1}} {x_ {2} -x_ {1} }} (x -x_ {1}), \,}$

where ₁ , y _{1 2} 2 2 2 2 ) are two points in the line with x ₂ ? x ₁ . This is equivalent to the above point-slope form, where the slope is explicitly given as ( y ₂ Ã,/sub>)/( x ₂ Ã, - x ₁ ).

Mengalikan kedua sisi persamaan ini dengan ( x ₂ Â- x ₁ ) menghasilkan bentuk garis biasanya disebut sebagai bentuk simetris :

${\ displaystyle (x_ {2} -x_ {1}) (y-y_ {1}) = (y_ {2} -y_ {1}) (x-x_ {1}). \,}  ÂÂ$

Memperluas produk dan menyusun kembali istilah mengarah ke bentuk umum:

${\ displaystyle x \, (y_ {2} -y_ {1}) - y \, (x_ {2} -x_ {1}) = x_ {1} y_ {2} -x_ {2} y_ {1}}  ÂÂ$

Menggunakan determinan, seseorang mendapat bentuk penentu , mudah diingat:

${\ displaystyle {\ begin {vmatrix} x & amp; y & amp; 1 \\ x_ {1} & amp; y_ {1} & amp; 1 \\ x_ {2} & amp; y_ {2} & amp; 1 \ end {vmatrix}} = 0 \ ,.}  ÂÂ$

Intercept form

${\ displaystyle {\ frac {x} {a}} {\ frac {y} {b}} = 1, \,}  ÂÂ$

where a and b should be zero. The equation graph has x -with a and y -accept b . Intercept shapes in standard form with A / C = 1/ a and B i> = 1/ b . Lines that pass origin or horizontally or vertically violate non-zero conditions at a or b and can not be represented in this form.

Matrix form

Menggunakan urutan bentuk standar

${\ displaystyle Ax = C, \,}  ÂÂ$

seseorang dapat menulis ulang persamaan dalam bentuk matriks:

${\ displaystyle {\ begin {pmatrix} A & amp; B \ end {pmatrix}} {\ begin {pmatrix} x \\ y \ end {pmatrix}} = {\ start {pmatrix} C \ end {pmatrix}}.}  ÂÂ$

Selanjutnya, representasi ini meluas ke sistem persamaan linear.

${\ displaystyle A_ {1} x B_ {1} y = C_ {1}, \,}  ÂÂ$

${\ displaystyle A_ {2} x B_ {2} y = C_ {2}, \,}  ÂÂ$

menjadi:

${\ displaystyle {\ begin {pmatrix} A_ {1} & amp; B_ {1} \\ A_ {2} & amp; B_ {2} \ end {pmatrix}} { \ begin {pmatrix} x \\ y \ end {pmatrix}} = {\ begin {pmatrix} C_ {1} \\ C_ {2} \ end {pmatrix}}.}  ÂÂ$

Because it extends easily to higher dimensions, it is a general representation in linear algebra, and in computer programming. There is a method named to solve a system of linear equations, such as Gauss-Jordan which can be expressed as a basic matrix line operation.

Parametric form

${\ displaystyle x = Tt U \,}$

dan

${\ displaystyle y = Vt W. \,}  ÂÂ$

Ini adalah dua persamaan simultan dalam hal parameter variabel t , dengan kemiringan m = V / T , x -menghadapi ( VU - WT )/ V dan y -menghadapi ( WT - VU )/ T . Ini juga dapat dikaitkan dengan bentuk dua titik, di mana T = p - h , U = h , V = q - k , dan W = k :

${\ displaystyle x$

Source of the article : Wikipedia