Math Examples

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These are examples. For more extensive help, see metawikipedia:Help:Displaying_a_formula.

a1x + b1y = c1
a2x + b2y = c2

a_{11}x_{1} + a_{12}x_{2} + \dots + a_{1n}x_{n} = b_{1}
a_{21}x_{1} + a_{22}x_{2} + \dots + a_{2n}x_{n} = b_{2}
\vdots
a_{m1}x_{1} + a_{m2}x_{2} + \dots + a_{mn}x_{n} = b_{m}


A=\left(
  \begin{array}{ccc}
   a_{1,1} & \ldots & a_{1,n} \\
   \vdots  & \ddots & \vdots \\
   a_{m,1} & \ldots & a_{m,n}
  \end{array}
 \right)
, 
\bar{x} = \left(
  \begin{array}{c}
   x_1 \\
   \vdots \\
   x_n
  \end{array}
 \right)
, and 
\bar{b}=\left(
  \begin{array}{c}
   b_1 \\
   \vdots \\
   b_n
  \end{array}
 \right)

\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n.

  • Practical example

\sum_{k=2}^6 k^2 = 2^2+3^2+4^2+5^2+6^2 = 90.

 \prod_{i=m}^{n} x_{i} := x_{m} \cdot x_{m+1} \cdot x_{m+2} \cdot \cdots \cdot x_{n-1} \cdot x_{n}.

  • Practical example

 \prod_{i=2}^{6} \left(1 + {1\over i}\right) = \left(1 + {1\over 2}\right) \cdot \left(1 + {1\over 3}\right) \cdot \left(1 + {1\over 4}\right) \cdot \left(1 + {1\over 5}\right) \cdot \left(1 + {1\over 6}\right) = {7\over 2}.

\bar{x}=A^{-1}\bar{b}

det(A)\neq 0

m = n

det(A) = 0

m > n


Examples of linear equation systems
x + 5y = 8 x1 + 2x2 − 3x3 = 14
s = 1 / 2x + 3t − 1
Examples of nonlinear equation systems
x + y2 = 0 a_{1}x_{1} + a_{2}x_{2} + \dots + a_{n}x_{n} = 1
s − sint = 0  \sqrt{x} + 3y + 9z = 2

[edit] More examples

x + 2y = 10,\,

3a + 472b = 10b + 37,\,

2x + y -5 = -7x + 4y +3.\,

Ax + By + C = 0,\,

y - k = \frac{q - k}{p - h} (x - h),

 y \sin \phi + x \cos \phi - p = 0,\,

a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b.

x = \frac{\sqrt{4ac+b^2}-b}{2a}

\left|x+\frac{b}{2a}\right| = \frac{\sqrt{b^2-4ac\  }}{|2a|}\Rightarrow x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac\  }}{2a}.

[edit] Matrix example

A = \begin{bmatrix}
9 & 8 & 6 \\
1 & 2 & 7 \\
4 & 9 & 2 \\
6 & 0 & 5 \end{bmatrix}   or   A = \begin{pmatrix}
9 & 8 & 6 \\
1 & 2 & 7 \\
4 & 9 & 2 \\
6 & 0 & 5 \end{pmatrix}

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